Coloring curves intersecting a fixed line - Semantic Scholar
COLORING CURVES INTERSECTING A FIXED LINE ALEXANDRE ROK AND BARTOSZ WALCZAK
arXiv:1512.06112v1 [math.CO] 18 Dec 2015
Abstract. Let F be a family of curves in the plane with the following properties: (1) each member of F intersects a fixed straight line L in at least one and at most t points, (2) any two members of F intersect in at most one point, (3) the intersection graph of F is triangle-free. We prove that the chromatic number χ(F) of the intersection graph of F is bounded by a function of t. Dependence on t is crucial—it follows easily from the existence of triangle-free segment intersection graphs with arbitrarily large chromatic number that χ(F) can be arbitrarily large as t grows. It has been conjectured that the intersection graphs of families of curves F satisfying just condition 1 have chromatic number bounded in terms of t and the clique number, which would generalize the recent result that the class of outerstring graphs is χ-bounded. We also show that it is enough to establish the case t = 2 in order to prove the conjecture for any t.
1. Introduction Context. A curve in the plane is a homeomorphic image of the real interval [0, 1]. A family of curves F is simple if any two curves from F intersect in at most one point. The chromatic number χ(F) and the clique number ω(F) of a family of curves F are the chromatic number and the clique number of the intersection graph of F, respectively. A family of curves F is triangle-free if ω(F) = 2. Combinatorial and algorithmic aspects of intersection graphs of curves in the plane, also known as string graphs, have been attracting researchers for decades. A significant part of this research has been devoted to understanding classes of string graphs that are χ-bounded, which means that every graph G in the class satisfies χ(G) 6 f (ω(G)) for some function f : N → N, where χ(G) and ω(G) denote the chromatic number and the clique number of G, respectively. Only recently it has been proved that the class of all string graphs does not have this property. Theorem 1.1 (Pawlik et al. [20]). There are triangle-free families of straight-line segments with arbitrarily large chromatic number. The construction from the proof of Theorem 1.1 requires a lot of freedom in placing the segments around in the plane. Hence, it is natural to ask what additional restrictions on the placement of curves lead to χ-bounded classes of intersection graphs. One possible restriction is that each curve must intersect a fixed line in exactly one point. This has been proposed by McGuinness [16, 17], who proved the following. Theorem 1.2 (McGuinness [17]). Triangle-free simple families of curves each intersecting a fixed line in exactly one point have bounded chromatic number. The most general result in this direction, due to the authors, is as follows. Theorem 1.3 ([21]). The class of intersection graphs of families of curves each intersecting a fixed line in exactly one point is χ-bounded. Alexandre Rok was partially supported by Israel Science Foundation grant 1136/12. 1
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Do the statements from Theorems 1.2 and 1.3 remain valid if we allow more than one intersection of a curve with the fixed line? In view of Theorem 1.1, we cannot hope for any similar result for families of curves intersecting a fixed line at least once but with no upper bound on the number of intersections, because they are as general as unrestricted families of curves. However, imposing an upper bound on the number of intersections may be enough to obtain an analogue of Theorem 1.3. Conjecture 1.4 ([21]). For every t > 1, the class of intersection graphs of families of curves each intersecting a fixed line in at least one and at most t points is χ-bounded. Results. We present two result which offer the first progress on the way towards resolving Conjecture 1.4. Theorem 1.5. For every t > 2, triangle-free simple families of curves intersecting a fixed line in at least one and at most t points have bounded chromatic number. Theorem 1.6. For every t > 2, the following are equivalent: (1) The class of intersection graphs of families of curves each intersecting a fixed line in at least one and at most t points is χ-bounded. (2) The class of intersection graphs of families of curves each intersecting a fixed line in exactly two points is χ-bounded. Theorem 1.5 confirms Conjecture 1.4 for triangle-free simple families of curves, while Theorem 1.6 asserts that all the difficulty in Conjecture 1.4 lies in the case t = 2. Furthermore, the only bottleneck in the proof of Theorem 1.5 which prevents us from generalizing it to families of curves that are not necessarily simple is the use of a technical result of McGuinness [18] (see Lemma 4.2 in the present paper), which is proved for triangle-free simple families of curves and which remains unknown for more general families of curves. This also seems to be the main obstacle in generalizing Theorem 1.5 to higher clique number. The equivalence from Theorem 1.6 still holds after imposing some further restrictions on the family of curves in the second statement (see Lemma 3.5). Related work. The chromatic number of intersection graphs of geometric objects has been investigated since the seminal paper of Asplund and Gr¨ unbaum [2] from 1960. They proved that families of axis-parallel rectangles in the plane have chromatic number at most O(ω 2 ) and conjectured that for every d > 1, there is a function fd : N → N such that families of axis-parallel boxes in Rd have chromatic number at most fd (ω). However, a few years later a surprising construction due to Burling [4] showed that there are triangle-free families of axisparallel boxes in R3 with arbitrarily large chromatic number. Since then, the upper bound of O(ω 2 ) and the trivial lower bound of Ω(ω) on the maximum possible chromatic number of rectangle intersection graphs have only been improved in terms of multiplicative constants [8, 10]. Another classical example of a χ-bounded class of geometric intersection graphs is provided by interval overlap graphs—graphs defined on families of intervals in which two intervals are connected by an edge whenever they intersect but are not nested. Gy´arf´as [6, 7] proved that interval overlap graphs have chromatic number O(ω 2 4ω ). This bound was subsequently improved to O(ω 2 2ω ) by Kostochka [9] and then to O(2ω ) by Kostochka and Kratochv´ıl [11]. The best construction of interval overlap graphs with large chromatic number forces Ω(ω log ω) colors [9]. McGuinness [16, 17] proposed to investigate the problem from a different perspective—to allow much more general geometric shapes, but to restrict the way how they can be positioned
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in the plane. First, he proved that the class of intersection graphs of L-shapes intersecting a fixed horizontal line is χ-bounded [16]. Then, he also showed that triangle-free simple families of curves each intersecting a fixed line in exactly one point have bounded chromatic number [17] (see Theorem 1.2 above). Further progress in this direction was made by Suk [22], who proved that simple families of x-monotone curves intersecting a fixed vertical line give rise to a χ-bounded class of graphs, and by Laso´ n et al. [14], who showed that the same holds without the assumption that the curves are x-monotone. Finally, Rok and Walczak [21] proved that the class of intersection graphs of curves intersecting a fixed line in exactly one point is χ-bounded (see Theorem 1.3 above). On the other hand, the class of string graphs is not χ-bounded. Pawlik et al. [19, 20] presented a construction of triangle-free intersection graphs of segments or geometric shapes of various other kinds with arbitrarily large chromatic number (see Theorem 1.1 above). It grows as fast as Θ(log log n) with the number of vertices n. For string graphs with higher clique numbers, a slightly better construction forcing Θω ((log log n)ω−1 ) colors was presented by Krawczyk and Walczak [13]. The best upper bound on the chromatic number of string graphs in terms of the number of vertices is (log n)O(log ω) proved by Fox and Pach [5] using a separator theorem for string graphs due to Matouˇsek [15]. For segments or, more generally, x-monotone curves, this can be improved to Oω (log n), which follows from the above-mentioned result of Suk [22] by a standard divide-and-conquer argument. Upper bounds of the form Oω ((log log n)f (ω) ) are known for the classes of rectangle overlap graphs and subtree overlap graphs [12, 13]. The former still allow the construction for χ = Θ(log log n) and the latter for χ = Θω ((log log)ω−1 ). A problem related to bounding the chromatic number of string graphs concerns so-called k-quasi-planar graphs, that is, graphs that can have a curvilinear drawing in the plane with no k pairwise crossing edges. A well-known conjecture asserts that k-quasi-planar graphs with n vertices have Ok (n) edges [3]. This is known to be true up to k = 4 due to Ackerman [1]. The best known general upper bound is n(log n)O(log k) due to Fox and Pach [5]. Suk and Walczak [23] proved the upper bound of Ok (n log n) for the case that any two curves intersect in at most c one point and 2α(n) n log n for the case that any two curves intersect in at most t points, where α is the inverse Ackermann function and c = c(k, t). Conjecture 1.4, if true, would yield the upper bound of Ok (n log n) also in the latter case.
2. Terminology and notation We fix a horizontal line and call it the baseline. This is the line that curves from the families that we consider are required to intersect. For any considered family of curves F, we assume that every intersection of a curve c ∈ F with the baseline is proper, which means that c passes from one to the other side of the line at that point, and we assume that no two curves from F intersect the baseline in a common point. The closed halfplanes above and below the baseline are denoted by H + and H − , respectively. A basepoint of a curve c is an intersection point of c with the baseline. The leftmost and rightmost intersection points of a curve c with the baseline are called the left basepoint and the right basepoint of c and are denoted by l(c) and r(c), respectively. For a curve c intersecting the baseline, by following c from either of its endpoints to the first intersection point with the baseline, we obtain two subcurves L(c) and R(c) such that the basepoint of L(c) is to the left of the basepoint of R(c). We call such subcurves singlecurves—L(c) is the left single-curve, and R(c) is the right single-curve of c. Single-curves are distinguished in our terminology to prevent confusion with the full curves from the families
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considered. For any single-curve s, we let b(s) denote the unique basepoint of s, which is always an endpoint of s. For a curve c intersecting the baseline, we let l(c) = b(L(c)) and r(c) = b(R(c)). We introduce the order < of the points on the baseline so that a < b when a is to the left of b. If c1 and c2 are two curves (or single-curves) intersecting the baseline, then c1 ≺ c2 denotes that all basepoints of c1 lie to the left of all basepoints of c2 . This notation extends naturally to families of curves (or single-curves): F1 ≺ F2 if c1 ≺ c2 for any c1 ∈ F1 and c2 ∈ F2 . An interval is a segment of the baseline. If a and b are two points on the baseline such that a < b, then [a, b] and (a, b) denote the closed interval and the open interval, respectively, with endpoints a and b. If F is a family a curves (or single-curves) intersecting the baseline, then F(a, b) denotes the curves (or single-curves) from F whose all basepoints belong to the interval (a, b). 3. Curves intersecting the baseline a bounded number of times The goal of this section is to establish Theorem 1.6 and its variant, Lemma 3.6, which is essential to the proof of theorem 1.5 presented in the next section. A restricted curve is a curve c intersecting the baseline in at least two points such that L(c), R(c) ⊂ H + . It follows that every restricted curve intersects the baseline in an even number of points. A family of curves F is restricted if every curve in F is restricted and any intersection between two curves c1 , c2 ∈ F is either between R(c1 ) and L(c2 ) or between L(c1 ) and R(c2 ). Lemma 3.1. Let ft−1 : N → N be a nondecreasing function such that every family G of curves intersecting the baseline in at least one and at most t − 1 points satisfies χ(G) 6 ft−1 (ω(G)), where t > 2. Let F be a family of curves intersecting the baseline in at least one and at most t points. 2 (ω(F)). (1) If t is odd, then χ(F) 6 ft−1 2 (ω(F)) such that any (2) If t is even, then there is a subfamily G ⊂ F with χ(G) > χ(F)/ft−1 intersection between two curves c1 , c2 ∈ G is either between R(c1 ) and L(c2 ) or between L(c1 ) and R(c2 ).
Proof. For a curve c ∈ F and a single-curve s ∈ {L(c), R(c)}, let c ⊖ s denote the part of c that remains after s and a very small open neighborhood of b(s) disjoint from all other curves in F have been removed. For a curve c ∈ F intersecting the baseline in an odd number of points, let U (c) and D(c) denote the upper single-curve of c (i.e. the one contained in H + ) and the lower single-curve of c (i.e. the one contained in H − ), respectively. Transform F as follows to obtain two families F1 and F2 intersecting the baseline in at least one and at most t points, For every curve c ∈ F intersecting the baseline in exactly t points, • if t is odd, then put c ⊖ U (c) to F1 and c ⊖ D(c) to F2 . • if t is even, then put c ⊖ L(c) to F1 and c ⊖ R(c) to F2 , Furthermore, put every curve from F intersecting the baseline in fewer than t points to both F1 and F2 . It follows that χ(F1 ) 6 ft−1 (ω(F1 )) 6 ft−1 (ω(F)) and χ(F2 ) 6 ft−1 (ω(F2 )) 6 ft−1 (ω(F)). For every curve c ∈ F and for k ∈ {1, 2}, let φk (c) be the color of the curve obtained from c in a proper coloring of Fk with χ(Fk ) 6 ft−1 (ω(F)) colors. Any two curves c1 , c2 ∈ F such that (φ1 (c1 ), φ2 (c1 )) = (φ1 (c2 ), φ2 (c2 )) can intersect only when their opposite single-curves (as identified above) intersect. Therefore, if t is odd, then the coloring (φ1 , φ2 ) of F is proper, because an upper single-curve and a lower single-curve cannot intersect. If t 2 (ω(F)) such that the pair of is even, then there is a subfamily G ⊂ F with χ(G) > χ(F)/ft−1 colors (φ1 (c), φ2 (c)) is constant over all c ∈ G, that is G is as required in the lemma.
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A section of a restricted curve c is a part of c between two intersection points with the baseline that are consecutive along c. A section s of a restricted curve c is a cap of c if s ⊂ H + . It follows that a restricted curve intersecting the baseline in 2n points determines n − 1 caps. Two single-curves p, q ∈ H + are separated by a cap s if p and q lie entirely in two different arc-connected components of H + r s. Two single-curves p, q ∈ H + are separated in a restricted family F if they are separated by some cap of a curve from F. Lemma 3.2. Let F be a restricted family of curves, and let G ⊆ F. If L(c) and R(c) are separated in F for every curve c ∈ G, then χ(G) 6 4. Proof. Let S = {L(c) : c ∈ G} ∪ {R(c) : c ∈ G}. Construct an auxiliary graph G as follows. The S vertices of G are the arc-connected components of S. The edges of G are the pairs of vertices U V for which there is c ∈ G such that L(c) ⊆ U and R(c) ⊆ V or vice versa. Suppose G has a loop at a vertex V . Then there is c ∈ G such that L(c), R(c) ⊂ V . Since L(c) and R(c) are separated in F, there is a cap s of a curve in F such that L(c) and R(c) are separated by s. Therefore, by the Jordan curve theorem, V and s must intersect. This and the S fact that V ⊆ F contradict the assumption that F is restricted. Hence, G has no loops. The graph G is planar. Indeed, each curve c ∈ G such that L(c) ⊆ U and R(c) ⊆ V , witnessing an edge (U, V ) of G, connects U with V in such a way that there are no intersections with other curves in between. This is because any such intersection would contradict the assumption that F is restricted. Thus χ(G) 6 4. Fix a proper 4-coloring of G, and let the color of a curve c ∈ G be defined as the color of the vertex of G containing L(c). This yields a proper 4-coloring of G. Indeed, if two curves c1 , c2 ∈ G, then either R(c1 ) intersects L(c2 ) or L(c1 ) intersects R(c2 ), so the two vertices of G after which c1 and c2 inherit their colors are connected by an edge of G, witnessed by c1 if R(c1 ) intersects L(c2 ) or by c2 if L(c1 ) intersects R(c2 ). Lemma 3.3. Let F be a restricted family of curves, and let c1 , c2 ∈ F. If neither L(c1 ) and R(c1 ) nor L(c2 ) and R(c2 ) are separated in F, then the intervals [l(c1 ), r(c1 )] and [l(c2 ), r(c2 )] are nested or disjoint. Proof. Suppose to the contrary that [l(c1 ), r(c1 )] and [l(c2 ), r(c2 )] are neither nested nor disjoint. Then l(c1 ) < l(c2 ) < r(c1 ) < r(c2 ) or l(c2 ) < l(c1 ) < r(c2 ) < r(c1 ). Assume the former, without loss of generality. Connect the endpoints of c2 by a curve p lying strictly above the baseline and avoiding all other points of c2 . Such a curve exists, otherwise L(c2 ) and R(c2 ) would be separated by a cap of c2 . Let C denote the closed curve obtained by taking the union of p and c2 . For every cap s of c2 , since L(c1 ) and R(c1 ) are not separated by s, the segment [l(c1 ), r(c1 )] contains either both or neither of the endpoints of s. It also contains l(c2 ) but not r(c2 ). It follows that the number of intersections of c2 with the segment [l(c1 ), r(c1 )] is odd. Therefore, by the Jordan curve theorem, l(c1 ) and r(c1 ) lie in two different regions of C. Let x and y be two intersection points of c1 with the baseline such that x is in the same region of C as l(c1 ), y is in the same region of C as r(c1 ), and x, y are consecutive on c1 . Let s1 denote the section of c1 between x and y. It does not intersect c2 , because c2 can only intersect the parts L(c1 ) and R(c1 ) of c1 . Clearly, s1 and C must intersect, which is therefore possible only when s1 and p intersect. This implies that s1 ⊂ H + , because p ⊂ H + . The segment [x, y] intersects C and thus c2 an odd number of times. For every cap s2 of c2 , the segment [x, y] contains either both or neither of the endpoints of s2 , because s1 does not intersect c2 . Therefore, [x, y] contains exactly one of the points l(c2 ), r(c2 ). This means that L(c2 ) and R(c2 ) are separated by s1 , which is a contradiction.
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Lemma 3.4. Suppose there is a function f : N → N such that every restricted family F˜ of curves intersecting the baseline in exactly two points satisfies χ(F˜ ) 6 f (ω(F˜ )). Then every restricted family of curves F satisfies χ(F) 6 f (ω(F)) + 4. Proof. Let F be a restricted family of curves. Let G ⊆ F be the family of those curves c ∈ F for which L(c) and R(c) are separated in F. By Lemma 3.2, χ(G) 6 4. Let H = F r G. By Lemma 3.3, for any c1 , c2 ∈ H, the intervals [l(c1 ), r(c1 )] and [l(c2 ), r(c2 )] are nested or disjoint, and therefore the semicircles in H − with diameters [l(c1 ), r(c1 )] and [l(c2 ), r(c2 )] are disjoint. Transform each curve c ∈ H replacing its part between l(c) and r(c) by a semicircle ˜ with the same with diameter [l(c), r(c)] contained in H − . This way a restricted family H ˜ intersects the baseline in exactly two intersection graph as H is obtained. Every curve in H ˜ 6 f (ω(H)) ˜ = f (ω(H)). We points. Therefore, by the assumption of the lemma, χ(H) = χ(H) conclude that χ(F) 6 χ(G) + χ(H) 6 f (ω(H)) + 4. Lemma 3.5. Suppose there is a (nondecreasing) function f : N → N such that every restricted ˜ 6 f (ω(F)). ˜ family F˜ of curves intersecting the baseline in exactly two points satisfies χ(F) Let f1 : N → N be the function claimed by Theorem 1.3, that is, such that every family H of curves intersecting the baseline in exactly one point satisfies χ(H) 6 f1 (ω(H)). For t > 2, let ( 2 (x) if t is odd, ft−1 ft (x) = 2 (f (x) + 4)ft−1 (x) if t is even. Then, for t > 1, every family F of curves intersecting the baseline in at least one and at most t points satisfies χ(F) 6 ft (ω(F)). Proof. The proof goes by induction on t. There is nothing to prove for t = 1 (the conclusion repeats an assumption), so suppose t > 2. Let F be a family of curves intersecting the baseline in at least one and at most t points. If t is odd, then it follows from Lemma 3.1 that χ(F) 6 2 (ω(F)) = f (ω(F)). Now, suppose t is even. By Lemma 3.1, there is a subfamily G ⊂ F ft−1 t 2 (ω(F)) such that any intersection between two curves c , c ∈ G is with χ(G) > χ(F)/ft−1 1 2 either between R(c1 ) and L(c2 ) or between L(c1 ) and R(c2 ). Since the curves c ∈ G with L(c), R(c) ∈ H + cannot intersect the curves c ∈ G with L(c), R(c) ∈ H − and by symmetry, we can assume without loss of generality that L(c), R(c) ∈ H + for every c ∈ G. This implies that G is restricted and therefore, by Lemma 3.4, χ(G) 6 f (ω(G)) + 4 6 f (ω(F)) + 4. We conclude 2 (ω(F)) 6 (f (ω(G)) + 4)f 2 (ω(F)) as required. that χ(F) 6 χ(G)ft−1 t−1 Theorem 1.6 is now a direct consequence of Lemma 3.5. The exact same argument as in Lemmas 3.1–3.5 but for triangle-free families of simple curves intersecting a fixed line in at least one and at most t points gives the following reduction. Lemma 3.6. Suppose there is a constant α such that every triangle-free simple restricted family F˜ of curves intersecting the baseline in exactly two points satisfies χ(F˜ ) 6 α. Let β1 be the constant claimed by Theorem 1.2, that is, such that every triangle-free simple family H of curves intersecting the baseline in exactly one point satisfies χ(H) 6 β1 . For t > 2, let ( 2 if t is odd, βt−1 βt (x) = 2 (α + 4)βt−1 if t is even. Then, for t > 1, every triangle-free simple restricted family F of curves intersecting the baseline in at least one and at most t points satisfies χ(F) 6 βt .
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s2 s 1 p1
p2 p3
p4
Figure 1. A family {p1 , p2 , p3 , p4 } externally supported by s1 and s2 . 4. Curves intersecting a fixed line exactly twice This section is devoted to the proof of Theorem 1.5. 4.1. Preliminaries. The exterior of a family of curves F, denoted by ext(F), is the unique S unbounded arc-connected component of R2 r G. A family G ⊂ F is externally supported in F if for every curve p ∈ G there is a curve s ∈ F that intersects p and ext(G). Such an s is called the support of p when the family G is implicit. The following lemma is a straightforward application of the ideas expressed by Gy´arf´as [6, 7], which were subsequently used in [16, 17, 22, 14, 21]. Lemma 4.1. Every family of curves F with χ(F) > 2 has a subfamily G that is externally supported in F and satisfies χ(G) > χ(F)/2. Proof. We can assume without loss of generality that the intersection graph of F is connected, otherwise we can restrict ourselves to the connected component with maximum chromatic number. Let c0 be an arbitrary curve in F that intersects the boundary of ext(F) at a point that does not belong to any other curve in F. For i > 0, let Fi denote the family of curves in F that are at distance i from c0 in the intersection graph of F. It follows that F0 = {c0 }, S S S F = i>0 Fi = {c0 } ∪ F , and each curve in Fi is disjoint from each i>0 F2i+1 ∪ i>0 S 2i+2 S curve in Fj whenever |i − j| > 1. Thus χ( i>0 F2i+1 ) > χ(F)/2 or χ( i>0 F2i+2 ) > χ(F)/2, so there is d > 1 with χ(Fd ) > χ(F)/2. We claim that Fd is externally supported in F. Fix cd ∈ Fd , and let c0 . . . cd be a shortest path from c0 to cd in the intersection graph of F. Since c0 hits the boundary of ext(F), it S intersects ext(Fd ). Moreover, c0 , . . . , cd−2 are disjoint from Fd , as otherwise there would be a curve in Fd at distance less than d from c0 . Therefore, all c0 , . . . , cd−2 are entirely contained in ext(Fd ). This implies that cd−1 intersects ext(Fd ), so it is a support of cd . A closed curve is a homeomorphic image of a circle. By the Jordan curve theorem, for every closed curve C, the set R2 r C splits into two arc-connected components, one bounded, called the interior of C and denoted by int(C), and one unbounded, called the exterior of C and denoted by ext(C). The following is a special case of the main result proved in [18]. Lemma 4.2 (McGuinness [18]). There are constants B and K with the following property. Let F be a family of curves such that any two curves in F intersect in at most one point and the intersection graph of F is triangle-free. Then there is a closed curve C constained in the union
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of at most 5 curves from F such that if G ⊂ F is the family of curves from G contained in int(C), then χ(G) > χ(F)/K. 4.2. General setting. Lemma 3.6 reduces Theorem 1.5 to triangle-free simple restricted families of curves F with exactly two intersection points with the baseline. We fix this setting for the remainder of Section 4. Namely, we let F be a family of curves with the following properties: (1) Each curve in F intersects the baseline in exactly two points, both of which are proper intersections. (2) Both endpoints of each curve in F lie above the baseline. (3) Any intersection between two curves in F is an intersection between their single-curves of opposite types, that is, an intersection of a right single-curve with a left single-curve. (4) Any two curves in F intersect in at most one point. (5) The intersection graph of F is triangle-free. We remark that the only reason for the conditions 4 above is that it is required by Lemma 4.2. Proving an analogue of Lemma 4.2 for more general classes of curves (with any constant in place of 5) will automatically lead to a generalization of our Theorem 1.5. When s is a single-curve arising from a curve in F, then we write C(s) to denote the curve from F that gives rise to s. If S is a family of single-curves arising from curves in F, then we define C(S) = {C(s) : s ∈ S}. 4.3. Getting surrounded. If p1 < · · · < ps are the intersection points of a curve with the baseline, then we call the points pi and pi+1 consecutive for 1 6 i 6 s − 1. Note that this concept of being consecutive differs from the one used in Section 3. Lemma 4.3. Let C be a closed curve, and let a and b be two consecutive intersection points of C with the baseline such that (a, b) ⊂ int(C). If L, R ⊂ F are families of curves contained in int(C) such that only the left basepoint and only the right basepoint is in (a, b), respectively, then χ(L) 6 1 and χ(R) 6 1. Proof. We prove only that χ(L) 6 1, since the other case is analogous. Let c1 and c2 be two intersecting curves of L. If b′ is the next intersection point of C with the baseline after b, then for both c1 and c2 the interval between their basepoints must contain (b, b′ ). Without loss of generality L(c1 ) ≺ L(c2 ). Since no intersection between c1 and c2 can occur below the baseline, it follows that R(c2 ) ≺ R(c1 ). If L(c1 ) intersects R(c2 ) and p denotes their first intersection point as going along L(c1 ) from the basepoint, then the subcurve of L(c1 ) from the basepoint to p, the subcurve of c2 from p to the right basepoint of c2 , the interval [b(L(c1 )), b(L(c2 ))] and the subcurve of c2 connecting b(L(c2 )) and b(R(c2 )) form a closed curve contained in int(C) that contains the segment (b, b′ ) in its interior. This implies (b, b′ ) ⊂ int(C), contradicting the assumption that (a, b) ⊂ int(C). The case that R(c1 ) intersects L(c2 ) is analogous. Lemma 4.4. Let C be a closed curve contained in the union of at most 5 curves from F. If G ⊂ F and every curve in G is contained in int(C), then there are two consecutive intersection points a, b of C with the baseline such that χ(G(a, b)) > χ(G)/5 − 1. Proof. Let p1 < · · · < p2n be the intersection points of C with the baseline. For 1 6 k 6 n, let Gk denote the curves from G whose left basepoint is in (p2k+1 , p2k+2 ). Since C is contained in the union of at most 5 curves intersecting the baseline twice, we have n 6 5. Therefore, for some k ∈ {1, . . . , n}, we have χ(Gk ) > χ(G)/5. By Lemma 4.3, the curves from Gk whose right basepoint is not in (p2k+1 , p2k+2 ) have chromatic number at most 1. The remaining curves form the family G(p2k+1 , p2k+2 ), which has chromatic number at least χ(G)/5 − 1.
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p4
Figure 2. A skeleton of type 1 supporting some family {p1 , p2 , p3 }, while p4 is not supported. 4.4. Rising of skeletons. A skeleton is a pair (C, S), where C is a closed curve and S is a family of single-curves of the same type (all left or all right) arising from curves in F such that the basepoints of the single-curves in S belong to int(C). Since all members of S in a skeleton (C, S) are of the same type, they are pairwise disjoint. The initial part of a single-curve s ∈ S with respect to a skeleton (C, S) is the part of s between b(s) and the first intersection point of s with C. The initial part of s is denoted by s′ when the skeleton is clear from the context. A family G ⊂ F is supported by the skeleton (C, S) if every curve in G is contained in int(C) and intersects s′ for some s ∈ S. See Figure 2. Lemma 4.5. Let gα (x) = 2K(max{(20x + 5)K, B} + 2α), where K and B are the constants from Lemma 4.2. Let F˜ ⊂ F. If χ(F˜ ) > gα (x), then at least one of the following holds: (1) There is a subfamily G ⊂ F˜ with χ(G) > α supported by a skeleton (C, S) such that C is ˜ contained in the union of at most 5 curves from F˜ and S ⊂ F. ˜ (2) There is a subfamily P ⊂ F with χ(G) > x and a family of single-curves Y of common type arising from some curves from F˜ such that each p ∈ P intersects some s ∈ Y and P ≺ Y or Y ≺ P. Proof. Suppose the condition 1 does not hold. We apply Lemma 4.1 in order to get an externally ˜ supported subfamilly G ⊂ F˜ with χ(G) > χ(F)/2 > K(max{(20x + 5)K, B} + 2α). Next, we apply Lemma 4.2 to G to get a closed curve C contained in the union of at most 5 curves from ˜ > χ(G)/K > G such that the family G˜ ⊂ G of curves from G contained in int(C) satisfies χ(G) max{(20x + 5)K, B} + 2α. Let S denote the set of supports of G, that is, the set of curves from F˜ intersecting some curve from G and ext(G). Let S1 ⊂ L(S) and S2 ⊂ R(S) be the sets of those curves of L(S) and R(S), respectively, that are based in int(C). Every curve in S1 or S2 S intersects ext(C), because it intersects ext(G) and C ⊂ G. After removing from G˜ all curves supported by at least one of the skeletons (C, S1 ), (C, S2 ), we are left with a family H ⊂ G˜
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ALEXANDRE ROK AND BARTOSZ WALCZAK
˜ − 2α > max{(20x + 5)K, B}. Apply Lemma 4.2 again to H to get a such that χ(H) > χ(G) S ˜ ˜ of curves from H contained in int(C) ˜ satisfies closed curve C ⊂ H such that the family H ˜ > χ(H)/K > 20x + 5. By Lemma 4.4, there are two consective intersection points a and χ(H) ˜ and χ(H(a, ˜ b)) > χ(H)/5 ˜ b of C˜ with the baseline such that (a, b) ⊂ int(C) − 1 > 4x. ˜ b). If one basepoint of s lies in int(C) ˜ and Suppose s ∈ S is a support of a curve c ∈ H(a, ˜ ˜ the other in ext(C), then s must intersect C below the baseline, which is impossible in our ˜ setting. Suppose both basepoints of s lie in int(C). Then, since no intersection can occur ˜ However, below the baseline, the whole subcurve of s connecting l(s) and r(s) lies in int(C). ˜ as well. This implies, by the Jordan by assumption, s must intersect ext(C) and hence ext(C) curve theorem, that at least one of the single-curves of s intersects C˜ and then C as going along S the single-curve from the basepoint. In particular s ∈ Sk for some k ∈ {1, 2}. Since C˜ ⊂ H, the intersection point of that single-curve of s with C˜ belongs to a curve p ∈ H, as well as it
belongs to the initial part s′ of s with respect to the skeleton (C, Sk ). Therefore, p is supported by (C, Sk ), which contradicts the definition of H. This shows that every support of a curve in ˜ b) has both bases in ext(C). ˜ H(a, ˜ ˜ b) The family H(a, b) is the union of B1 and B2 , where B1 is the set of all curves c ∈ H(a, having a support s ∈ S such that L(s) intersects c and symmetrically for B2 with R(s). Clearly, ˜ b))/2 > 2x. Let S˜k be the family of single-curves for some k ∈ {1, 2}, we have χ(Bk ) > χ(H(a, of type k arising from the corresponding supports. Now Bk is the union of two sets A1 and A2 , where A1 is the set of all curves of Bk that are intersected by some single-curve of S˜k based to the left of Bk and symmetrically for A2 . For some j ∈ {1, 2}, we have χ(Aj ) > χ(Bk )/2 > x. Let P = Aj and Y be the subset of S˜k corresponding to Aj . It follows that the condition 2 is satisfied for P and Y. For a function h : R → R, we let h(m) denote the m-fold composition of h. (3) Lemma 4.6. Let f (α) = gα (4), where g is the function from Lemma 4.5. Let F˜ ⊂ F. If χ(F˜ ) > f (α), then there is a subfamily G ⊂ F˜ with χ(G) > α supported by a skeleton (C, S) such that C is contained in the union of at most 5 curves from F˜ and S ⊂ F˜ .
˜ For 1 6 i 6 3, let Fi Proof. Suppose that such a subfamily G does not exist. Let F0 = F. and Yi be the families obtained by applying the conclusion 2 of Lemma 4.5 to Fi−1 . That is, (3−i) Fi ⊂ Fi−1 , χ(Fi ) > gα (4), Yi is a family of single-curves of common type arising from some curves in Fi−1 , each c ∈ Fi intersects some s ∈ Yi , and Fi ≺ Yi or Yi ≺ Fi . Two of the Yi s, say Yi1 and Yi2 where 1 6 i1 < i2 6 3, must contain single-curves of the same type. We restrict ourselves only to the case that Yi2 ≺ Fi2 , as the opposite one is symmetric. Let X be the family of single-curves of the curves in Fi2 with the type opposite to that of Yi2 , and let Y = Yi2 . Process the single-curves in X in the order of their bases from right to left, adding them to a (3−i ) family H until χ(C(H)) = 2. Let D = X r H, so that χ(C(D)) > gα 2 (4) − 2 > 2. For c ∈ H, let p(c) be the first intersection point of c with a single-curve of Y encountered as going from b(c) along c, let s(c) be the single-curve of Y that intersects c at p(c), and let A(c) be the region enclosed in the closed curve formed by the subcurve of c (denoted by c˜) from b(c) to p(c), the subcurve of s(c) (denoted by s˜(c)) from b(s(c)) to p(c), and the part of the baseline between b(c) and b(s(c)). Let c be the single-curve in H with leftmost base. We claim that A(c) ⊂ A(c′ ) for any other single-curve in H. Indeed, it this is not true, then the boundary curve of A(c′ ) must hit A(c) at some point q when following counterclockwise starting from b(c′ ). Since the pairs of curves c, c′ and s(c), s(c′ ) have common types, they cannot intersect, so q is the intersection of c˜ and s(c′ ) or c˜′ and s(c). It follows that q = p(c) in the former and q = p(c′ ) in the latter case. In both cases s(c′ ) = s(c), so the boundary curve of A(c′ ) cannot enter the interior of A(c).
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˜ such that After removing from D all single-curves intersecting s˜(c), we obtain a family D S˜ ˜ D ⊂ A(c) and χ(C(D)) > χ(C(D)) − 1 > 1. Indeed, since the family F is triangle-free, the set of curves corresponding to the single-curves removed from D must be independent. Now, let ˜ and take any single-curve si from Yi intersecting v. For each single-curve c′ ∈ H, since H v∈D 1 1 and Y are single-curves of curves in Fi1 , si1 ∈ Yi1 , Fi1 ≺ Yi1 or Yi1 ≺ Fi1 , and v ⊂ A(c) ⊂ A(c′ ), the single-curve si1 must intersect c˜′ or s˜(c′ ). It cannot intersect s˜(c′ ), because s(c′ ) and si1 have the same type. Hence si1 intersects all single-curves in H. This yields χ(C(H)) 6 1, which is a contradiction. 4.5. Skeletons rule the world. To complete the proof of Theorem 1.5, we make use of families supported by skeletons to prove that if the chromatic number of F is too large, then it contains the intersection graph of F constains a triangle. If C is a closed curve intersecting the baseline, then an upper section of C is an arc-connected component of H + r C. Lemma 4.7. Let f˜(x) = f (5x), where f is the function defined in Lemma 4.6. Then χ(F) < f˜(11) (511 K(max{20K, B} + 11)), where K and B are the constants from Lemma 4.2. Proof. Suppose to the contrary that χ(F) > f˜(11) (511 K(max{20K, B} + 11)). Let F0 = F. For 1 6 i 6 11, proceed as follows. Apply Lemma 4.6 to Fi−1 to get a subfamily F˜i ⊂ Fi−1 with χ(F˜i ) > 5f˜(11−i) (511 K(max{20K, B} + 11)) supported by a skeleton (Ci , S˜i ), where Ci is a closed curve contained in the union of at most 5 curves from Fi−1 . Let p1 < · · · < p2n be the intersection points of Ci with the baseline. For 1 6 j 6 n, let Sij = S˜i (p2j−1 , p2j ), and let Fij consist of the curves from F˜i that are supported by the skeleton (Ci , Sij ). Thus S S S˜i = nj=1 Sij and F˜i = nj=1 Fij . Since n 6 5, there is j ∈ {1, . . . , n} such that χ(Fij ) > χ(F˜i )/5 > f˜(11−i) (511 K(max{20K, B} + 11)). Let Fi = F j and Si = S j . i
i
Apply Lemma 4.2 to F11 to get a closed curve C contained in the union of r 6 5 curves c1 , . . . , cr ∈ F11 and the subfamily G0 ⊂ F11 of curves contained in int(C) such that χ(G0 ) > χ(F11 )/K > 511 (max{20K, B} + 14). The part of C above the baseline is contained in the union of 2r 6 10 single-curves h1 , . . . , h2r (r of either type) arising from c1 , . . . , cr . Then, for 1 6 i 6 11, proceed as follows. For 1 6 j 6 2r, let Uij be the family of supports s ∈ Si such that the initial part s′ of s with respect to the skeleton (Ci , Si ) intersects hj , and let Gij be the set S S2r j j of curves from Gi−1 that have a support in Uij . Thus Si = 2r j=1 Ui and Gi−1 = j=1 Gi . Since r 6 5 and the supports in Si can only intersect single-curves of h1 , . . . , h2r of the opposite type, at most 5 of the families Uij and thus Gij are nonempty. Therefore, there is j ∈ {1, . . . , 2r} such that χ(Gij ) > χ(Gi−1 )/5 > 511−i (max{20K, B} + 11). Let Gi = Gij and Ui = Uij . Since 2r 6 10, there are two indices i1 and i2 with 1 6 i1 < i2 6 11 such that Ui1 = Uij1 and Ui2 = Uij2 for a common index j ∈ {1, . . . , 2r}. In particular, Ui1 and Ui2 are of the same type, which is opposite to the type of hj . Let h = hj . Let s1 ≺ · · · ≺ sm be the single-curves in Ui1 . Let p1 < · · · < p2n be the intersection points of Ci1 with the baseline. Let U be the upper section of int(Ci1 ) containing (p2j−1 , p2j ). Recall that there is j ∈ {1, . . . , n} such that b(s1 ), . . . , b(sm ) ∈ (p2j−1 , p2j ). The initial parts s′1 , . . . , s′m of s1 , . . . , sm with respect to the skeleton (Ci1 , Ui1 ) are contained in U (except for the endpoints of s′1 , . . . , s′m other than their basepoints). Since s′1 , . . . , s′m are pairwise disjoint, the orders of the endpoints of s′1 , . . . , s′m along the two parts of the boundary of U between p2j−1 and p2j are the same. Therefore, the set U r (s′1 ∪ · · · ∪ s′m ) splits into m + 1 arc-connected components U0 , . . . , Um such that • the boundary of U0 is formed by s′1 , [p2j−1 , b(s1 )], and the clockwise part of the boundary of U from p2j−1 to the other endpoint of s′1 ;
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ALEXANDRE ROK AND BARTOSZ WALCZAK
• for 1 6 k 6 m−1, the boundary of Uk is formed by s′k , s′k+1 , [b(sk ), b(sk+1 )], and the clockwise part of the boundary of U from the other endpoint of s′k to the other endpoint of s′k+1 ; • the boundary of Um is formed by s′m , [b(sm ), p2j ], and the clockwise part of the boundary of U from the other endpoint of s′m to p2j . Let X be the family of single-curves of the type opposite to s1 , . . . , sm arising from the curves in G11 . Thus every single-curve in X intersects s′k for some k ∈ {1, . . . , m}. The above implies that • (p2j−1 , b(s1 )) ⊂ U0 , so every single-curve in X (p2j−1 , b(s1 )) intersects s′1 ; • for every i ∈ {1, . . . , n} r {j}, if X (p2i−1 , p2i ) 6= ∅, then (p2i−1 , p2i ) ⊂ U ; consequently, there is ki ∈ {0, . . . , m} such that (p2i−1 , p2i ) ⊂ Uki , so every single-curve in X (p2i−1 , p2i ) intersects s′ki or s′ki +1 ; • (b(sm ), p2j ) ⊂ Um , so every single-curve in X (b(sm ), p2j ) intersects s′m . ˜ be the curves from G11 that do not intersect any of s′ , s′m , and s′ , s′ Let H 1 ki +1 for i ∈ ki ˜ {1, . . . , n} r {j}. Since n 6 5, it follows that χ(H) > χ(G11 ) − 10 > max{20K, B} + 1 Now, let ˜ is formed by s′ , s′m , [b(s1 ), b(sm )], ˜ = s′ ∪ U1 ∪ s′ ∪ · · · ∪ Um−1 ∪ s′m , so that the boundary of U U 1
2
1
and the clockwise part of the boundary of U from the other endpoint of s′1 to the other endpoint ˜ are of s′m . The single-curves of the type opposite to s1 , . . . , sm arising from the curves in H ˜ ˜ ˜ contained in U . The other single-curves of the curves in H are either contained in U or disjoint ˜ . Those curves from H ˜ whose other single-curves are contained in U ˜ form the family from U ˜ ˜ have chromatic H = H(b(s1 ), b(sm )), while those whose other single-curves are disjoint from U ˜ − 1 > max{20K, B}. number at most 1. Hence χ(H) > χ(H) Apply Lemma 4.2 to H to get a closed curve C˜ contained in the union of at most 5 curves ˜ such that χ(D) > χ(H)/K > 20. from H and the subfamily D ⊂ H of curves contained in int(C) ˜ Let q1 < · · · < q2n be the intersection points of C with the baseline. By Lemma 4.4, there is j ∈ {1, . . . , n} such that χ(D(q2j−1 , q2j )) > χ(D)/5 − 1 > 3. By an argument analogous to the one above, it can be proved that the curves from D(q2j−1 , q2j ) supported by (Ci2 , Ui2 rUi2 (q2j−1 , q2j )) intersect two specific supports from Ui2 r Ui2 (q2j−1 , q2j ) and hence have chromatic number at most 2. Therefore, since χ(D(q2j−1 , q2j )) > 3, at least one curve from D(q2j−1 , q2j ) must be supported by (Ci2 , Ui2 (q2j−1 , q2j )), so Ui2 (q2j−1 , q2j ) 6= ∅. Let x ∈ Ui2 (q2j−1 , q2j ), and let x ˜ be the opposite single-curve of C(x). It follows from ˜ that b(˜ ˜ otherwise C(x) would intersect C˜ below the b(x) ∈ (q2j−1 , q2j ) ⊂ int(C) x) ∈ int(C), baseline, which is impossible. This implies b(˜ x) ∈ (b(s1 ), b(sm )), because all intersection points of C˜ with the baseline lie in (b(s1 ), b(sm )). Since C(x) ∈ Fi1 , there is s ∈ Si1 that supports C(x) in the skeleton (Ci1 , Si1 ). Since Si1 and x have the same type, s intersects x ˜. Recall that each single-curve in Ui1 or Ui2 intersects h. In particular, so do s1 , sm and x. Let H be the closed curve formed by the segment [b(s1 ), b(sm )] of the baseline, the part of s1 from b(s1 ) to the first intersection point with h, the part of sm from b(sm ) to the first intersection point with h, and the part of h between the two intersection points. It follows that ˜ have the same type, at least one of the following int(H) ⊂ int(C) ⊂ int(Ci1 ). Since h and x holds: (1) x ˜ intersects s1 , (2) x ˜ intersects sm , or (3) x ˜ intersects s at a point in int(H). In case (1) or (2), we can assume without loss of generality that s = s1 or s = sm , respectively. In case (3), s must go out of int(H) in order to intersect Ci1 , which is only possible by crossing h, because s, s1 and sm have the same type. In each of the three cases, C(h), C(s) and C(x) form a triangle in the intersection graph, which is a contradiction. Proof of Theorem 1.5. Lemma 4.7 shows that triangle-free simple restricted family of curves each intersecting the baseline in exactly two points have bounded chromatic number. Therefore, the assumption of Lemma 3.6 is satisfied, and Theorem 1.5 directly follows.
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References [1] Eyal Ackerman. On the maximum number of edges in topological graphs with no four pairwise crossing edges. Discrete Comput. Geom., 41(3):365–375, 2009. [2] Edgar Asplund and Branko Gr¨ unbaum. On a colouring problem. Math. Scand., 8:181–188, 1960. [3] Peter Brass, William Moser, and J´ anos Pach. Research Problems in Discrete Geometry. Springer, New York, 2005. [4] James P. Burling. On coloring problems of families of prototypes. PhD thesis, University of Colorado, Boulder, 1965. [5] Jacob Fox and J´ anos Pach. Applications of a new separator theorem for string graphs. Combin. Prob. Comput., 23(1):66–74, 2014. [6] Andr´ as Gy´ arf´ as. On the chromatic number of multiple interval graphs and overlap graphs. Discrete Math., 55(2):161–166, 1985. [7] Andr´ as Gy´ arf´ as. Corrigendum: On the chromatic number of multiple interval graphs and overlap graphs. Discrete Math., 62(3):333, 1986. [8] Clemens Hendler. Schranken f¨ ur F¨ arbungs- und Cliquen¨ uberdeckungszahl geometrisch repr¨ asentierbarer Graphen. Master’s thesis, Freie Universit¨ at Berlin, 1998. [9] Alexandr Kostochka. On upper bounds for the chromatic numbers of graphs. Trudy Inst. Mat., 10:204–226, 1988. [10] Alexandr Kostochka. Coloring intersection graphs of geometric figures with a given clique number. In J´ anos Pach, editor, Towards a Theory of Geometric Graphs, volume 342 of Contemp. Math., pages 127–138. Amer. Math. Soc., Providence, 2004. [11] Alexandr Kostochka and Jan Kratochv´ıl. Covering and coloring polygon-circle graphs. Discrete Math., 163(1– 3):299–305, 1997. [12] Tomasz Krawczyk, Arkadiusz Pawlik, and Bartosz Walczak. Coloring triangle-free rectangle overlap graphs with O(log log n) colors. Discrete Comput. Geom., 53(1):199–220, 2015. [13] Tomasz Krawczyk and Bartosz Walczak. On-line approach to off-line coloring problems on graphs with geometric representations. submitted, arXiv:1402.2437. [14] Michal Laso´ n, Piotr Micek, Arkadiusz Pawlik, and Bartosz Walczak. Coloring intersection graphs of arcconnected sets in the plane. Discrete Comput. Geom., 52(2):399–415, 2014. [15] Jiˇr´ı Matouˇsek. Near-optimal separators in string graphs. Combin. Prob. Comput., 23(1):135–139, 2014. [16] Sean McGuinness. On bounding the chromatic number of L-graphs. Discrete Math., 154(1–3):179–187, 1996. [17] Sean McGuinness. Colouring arcwise connected sets in the plane I. Graphs Combin., 16(4):429–439, 2000. [18] Sean McGuinness. Colouring arcwise connected sets in the plane II. Graphs Combin., 17(1):135–148, 2001. [19] Arkadiusz Pawlik, Jakub Kozik, Tomasz Krawczyk, Michal Laso´ n, Piotr Micek, William T. Trotter, and Bartosz Walczak. Triangle-free geometric intersection graphs with large chromatic number. Discrete Comput. Geom., 50(3):714–726, 2013. [20] Arkadiusz Pawlik, Jakub Kozik, Tomasz Krawczyk, Michal Laso´ n, Piotr Micek, William T. Trotter, and Bartosz Walczak. Triangle-free intersection graphs of line segments with large chromatic number. J. Combin. Theory Ser. B, 105:6–10, 2014. [21] Alexandre Rok and Bartosz Walczak. Outerstring graphs are χ-bounded. In Siu-Wing Cheng and Olivier Devillers, editors, 30th Annual Symposium on Computational Geometry (SoCG 2014), pages 136–143. ACM, New York, 2014. [22] Andrew Suk. Coloring intersection graphs of x-monotone curves in the plane. Combinatorica, 34(4):487–505, 2014. [23] Andrew Suk and Bartosz Walczak. New bounds on the maximum number of edges in k-quasi-planar graphs. Comput. Geom., 50:24–33, 2015.
(Alexandre Rok) Department of Mathematics, Ben-Gurion University of the Negev, Be’er Sheva 84105, Israel E-mail address: [email protected] (Bartosz Walczak) Theoretical Computer Science Department, Faculty of Mathematics and Computer Science, Jagiellonian University, Krak´ ow, Poland E-mail address: [email protected]
arXiv:1512.06112v1 [math.CO] 18 Dec 2015
Abstract. Let F be a family of curves in the plane with the following properties: (1) each member of F intersects a fixed straight line L in at least one and at most t points, (2) any two members of F intersect in at most one point, (3) the intersection graph of F is triangle-free. We prove that the chromatic number χ(F) of the intersection graph of F is bounded by a function of t. Dependence on t is crucial—it follows easily from the existence of triangle-free segment intersection graphs with arbitrarily large chromatic number that χ(F) can be arbitrarily large as t grows. It has been conjectured that the intersection graphs of families of curves F satisfying just condition 1 have chromatic number bounded in terms of t and the clique number, which would generalize the recent result that the class of outerstring graphs is χ-bounded. We also show that it is enough to establish the case t = 2 in order to prove the conjecture for any t.
1. Introduction Context. A curve in the plane is a homeomorphic image of the real interval [0, 1]. A family of curves F is simple if any two curves from F intersect in at most one point. The chromatic number χ(F) and the clique number ω(F) of a family of curves F are the chromatic number and the clique number of the intersection graph of F, respectively. A family of curves F is triangle-free if ω(F) = 2. Combinatorial and algorithmic aspects of intersection graphs of curves in the plane, also known as string graphs, have been attracting researchers for decades. A significant part of this research has been devoted to understanding classes of string graphs that are χ-bounded, which means that every graph G in the class satisfies χ(G) 6 f (ω(G)) for some function f : N → N, where χ(G) and ω(G) denote the chromatic number and the clique number of G, respectively. Only recently it has been proved that the class of all string graphs does not have this property. Theorem 1.1 (Pawlik et al. [20]). There are triangle-free families of straight-line segments with arbitrarily large chromatic number. The construction from the proof of Theorem 1.1 requires a lot of freedom in placing the segments around in the plane. Hence, it is natural to ask what additional restrictions on the placement of curves lead to χ-bounded classes of intersection graphs. One possible restriction is that each curve must intersect a fixed line in exactly one point. This has been proposed by McGuinness [16, 17], who proved the following. Theorem 1.2 (McGuinness [17]). Triangle-free simple families of curves each intersecting a fixed line in exactly one point have bounded chromatic number. The most general result in this direction, due to the authors, is as follows. Theorem 1.3 ([21]). The class of intersection graphs of families of curves each intersecting a fixed line in exactly one point is χ-bounded. Alexandre Rok was partially supported by Israel Science Foundation grant 1136/12. 1
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Do the statements from Theorems 1.2 and 1.3 remain valid if we allow more than one intersection of a curve with the fixed line? In view of Theorem 1.1, we cannot hope for any similar result for families of curves intersecting a fixed line at least once but with no upper bound on the number of intersections, because they are as general as unrestricted families of curves. However, imposing an upper bound on the number of intersections may be enough to obtain an analogue of Theorem 1.3. Conjecture 1.4 ([21]). For every t > 1, the class of intersection graphs of families of curves each intersecting a fixed line in at least one and at most t points is χ-bounded. Results. We present two result which offer the first progress on the way towards resolving Conjecture 1.4. Theorem 1.5. For every t > 2, triangle-free simple families of curves intersecting a fixed line in at least one and at most t points have bounded chromatic number. Theorem 1.6. For every t > 2, the following are equivalent: (1) The class of intersection graphs of families of curves each intersecting a fixed line in at least one and at most t points is χ-bounded. (2) The class of intersection graphs of families of curves each intersecting a fixed line in exactly two points is χ-bounded. Theorem 1.5 confirms Conjecture 1.4 for triangle-free simple families of curves, while Theorem 1.6 asserts that all the difficulty in Conjecture 1.4 lies in the case t = 2. Furthermore, the only bottleneck in the proof of Theorem 1.5 which prevents us from generalizing it to families of curves that are not necessarily simple is the use of a technical result of McGuinness [18] (see Lemma 4.2 in the present paper), which is proved for triangle-free simple families of curves and which remains unknown for more general families of curves. This also seems to be the main obstacle in generalizing Theorem 1.5 to higher clique number. The equivalence from Theorem 1.6 still holds after imposing some further restrictions on the family of curves in the second statement (see Lemma 3.5). Related work. The chromatic number of intersection graphs of geometric objects has been investigated since the seminal paper of Asplund and Gr¨ unbaum [2] from 1960. They proved that families of axis-parallel rectangles in the plane have chromatic number at most O(ω 2 ) and conjectured that for every d > 1, there is a function fd : N → N such that families of axis-parallel boxes in Rd have chromatic number at most fd (ω). However, a few years later a surprising construction due to Burling [4] showed that there are triangle-free families of axisparallel boxes in R3 with arbitrarily large chromatic number. Since then, the upper bound of O(ω 2 ) and the trivial lower bound of Ω(ω) on the maximum possible chromatic number of rectangle intersection graphs have only been improved in terms of multiplicative constants [8, 10]. Another classical example of a χ-bounded class of geometric intersection graphs is provided by interval overlap graphs—graphs defined on families of intervals in which two intervals are connected by an edge whenever they intersect but are not nested. Gy´arf´as [6, 7] proved that interval overlap graphs have chromatic number O(ω 2 4ω ). This bound was subsequently improved to O(ω 2 2ω ) by Kostochka [9] and then to O(2ω ) by Kostochka and Kratochv´ıl [11]. The best construction of interval overlap graphs with large chromatic number forces Ω(ω log ω) colors [9]. McGuinness [16, 17] proposed to investigate the problem from a different perspective—to allow much more general geometric shapes, but to restrict the way how they can be positioned
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in the plane. First, he proved that the class of intersection graphs of L-shapes intersecting a fixed horizontal line is χ-bounded [16]. Then, he also showed that triangle-free simple families of curves each intersecting a fixed line in exactly one point have bounded chromatic number [17] (see Theorem 1.2 above). Further progress in this direction was made by Suk [22], who proved that simple families of x-monotone curves intersecting a fixed vertical line give rise to a χ-bounded class of graphs, and by Laso´ n et al. [14], who showed that the same holds without the assumption that the curves are x-monotone. Finally, Rok and Walczak [21] proved that the class of intersection graphs of curves intersecting a fixed line in exactly one point is χ-bounded (see Theorem 1.3 above). On the other hand, the class of string graphs is not χ-bounded. Pawlik et al. [19, 20] presented a construction of triangle-free intersection graphs of segments or geometric shapes of various other kinds with arbitrarily large chromatic number (see Theorem 1.1 above). It grows as fast as Θ(log log n) with the number of vertices n. For string graphs with higher clique numbers, a slightly better construction forcing Θω ((log log n)ω−1 ) colors was presented by Krawczyk and Walczak [13]. The best upper bound on the chromatic number of string graphs in terms of the number of vertices is (log n)O(log ω) proved by Fox and Pach [5] using a separator theorem for string graphs due to Matouˇsek [15]. For segments or, more generally, x-monotone curves, this can be improved to Oω (log n), which follows from the above-mentioned result of Suk [22] by a standard divide-and-conquer argument. Upper bounds of the form Oω ((log log n)f (ω) ) are known for the classes of rectangle overlap graphs and subtree overlap graphs [12, 13]. The former still allow the construction for χ = Θ(log log n) and the latter for χ = Θω ((log log)ω−1 ). A problem related to bounding the chromatic number of string graphs concerns so-called k-quasi-planar graphs, that is, graphs that can have a curvilinear drawing in the plane with no k pairwise crossing edges. A well-known conjecture asserts that k-quasi-planar graphs with n vertices have Ok (n) edges [3]. This is known to be true up to k = 4 due to Ackerman [1]. The best known general upper bound is n(log n)O(log k) due to Fox and Pach [5]. Suk and Walczak [23] proved the upper bound of Ok (n log n) for the case that any two curves intersect in at most c one point and 2α(n) n log n for the case that any two curves intersect in at most t points, where α is the inverse Ackermann function and c = c(k, t). Conjecture 1.4, if true, would yield the upper bound of Ok (n log n) also in the latter case.
2. Terminology and notation We fix a horizontal line and call it the baseline. This is the line that curves from the families that we consider are required to intersect. For any considered family of curves F, we assume that every intersection of a curve c ∈ F with the baseline is proper, which means that c passes from one to the other side of the line at that point, and we assume that no two curves from F intersect the baseline in a common point. The closed halfplanes above and below the baseline are denoted by H + and H − , respectively. A basepoint of a curve c is an intersection point of c with the baseline. The leftmost and rightmost intersection points of a curve c with the baseline are called the left basepoint and the right basepoint of c and are denoted by l(c) and r(c), respectively. For a curve c intersecting the baseline, by following c from either of its endpoints to the first intersection point with the baseline, we obtain two subcurves L(c) and R(c) such that the basepoint of L(c) is to the left of the basepoint of R(c). We call such subcurves singlecurves—L(c) is the left single-curve, and R(c) is the right single-curve of c. Single-curves are distinguished in our terminology to prevent confusion with the full curves from the families
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considered. For any single-curve s, we let b(s) denote the unique basepoint of s, which is always an endpoint of s. For a curve c intersecting the baseline, we let l(c) = b(L(c)) and r(c) = b(R(c)). We introduce the order < of the points on the baseline so that a < b when a is to the left of b. If c1 and c2 are two curves (or single-curves) intersecting the baseline, then c1 ≺ c2 denotes that all basepoints of c1 lie to the left of all basepoints of c2 . This notation extends naturally to families of curves (or single-curves): F1 ≺ F2 if c1 ≺ c2 for any c1 ∈ F1 and c2 ∈ F2 . An interval is a segment of the baseline. If a and b are two points on the baseline such that a < b, then [a, b] and (a, b) denote the closed interval and the open interval, respectively, with endpoints a and b. If F is a family a curves (or single-curves) intersecting the baseline, then F(a, b) denotes the curves (or single-curves) from F whose all basepoints belong to the interval (a, b). 3. Curves intersecting the baseline a bounded number of times The goal of this section is to establish Theorem 1.6 and its variant, Lemma 3.6, which is essential to the proof of theorem 1.5 presented in the next section. A restricted curve is a curve c intersecting the baseline in at least two points such that L(c), R(c) ⊂ H + . It follows that every restricted curve intersects the baseline in an even number of points. A family of curves F is restricted if every curve in F is restricted and any intersection between two curves c1 , c2 ∈ F is either between R(c1 ) and L(c2 ) or between L(c1 ) and R(c2 ). Lemma 3.1. Let ft−1 : N → N be a nondecreasing function such that every family G of curves intersecting the baseline in at least one and at most t − 1 points satisfies χ(G) 6 ft−1 (ω(G)), where t > 2. Let F be a family of curves intersecting the baseline in at least one and at most t points. 2 (ω(F)). (1) If t is odd, then χ(F) 6 ft−1 2 (ω(F)) such that any (2) If t is even, then there is a subfamily G ⊂ F with χ(G) > χ(F)/ft−1 intersection between two curves c1 , c2 ∈ G is either between R(c1 ) and L(c2 ) or between L(c1 ) and R(c2 ).
Proof. For a curve c ∈ F and a single-curve s ∈ {L(c), R(c)}, let c ⊖ s denote the part of c that remains after s and a very small open neighborhood of b(s) disjoint from all other curves in F have been removed. For a curve c ∈ F intersecting the baseline in an odd number of points, let U (c) and D(c) denote the upper single-curve of c (i.e. the one contained in H + ) and the lower single-curve of c (i.e. the one contained in H − ), respectively. Transform F as follows to obtain two families F1 and F2 intersecting the baseline in at least one and at most t points, For every curve c ∈ F intersecting the baseline in exactly t points, • if t is odd, then put c ⊖ U (c) to F1 and c ⊖ D(c) to F2 . • if t is even, then put c ⊖ L(c) to F1 and c ⊖ R(c) to F2 , Furthermore, put every curve from F intersecting the baseline in fewer than t points to both F1 and F2 . It follows that χ(F1 ) 6 ft−1 (ω(F1 )) 6 ft−1 (ω(F)) and χ(F2 ) 6 ft−1 (ω(F2 )) 6 ft−1 (ω(F)). For every curve c ∈ F and for k ∈ {1, 2}, let φk (c) be the color of the curve obtained from c in a proper coloring of Fk with χ(Fk ) 6 ft−1 (ω(F)) colors. Any two curves c1 , c2 ∈ F such that (φ1 (c1 ), φ2 (c1 )) = (φ1 (c2 ), φ2 (c2 )) can intersect only when their opposite single-curves (as identified above) intersect. Therefore, if t is odd, then the coloring (φ1 , φ2 ) of F is proper, because an upper single-curve and a lower single-curve cannot intersect. If t 2 (ω(F)) such that the pair of is even, then there is a subfamily G ⊂ F with χ(G) > χ(F)/ft−1 colors (φ1 (c), φ2 (c)) is constant over all c ∈ G, that is G is as required in the lemma.
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A section of a restricted curve c is a part of c between two intersection points with the baseline that are consecutive along c. A section s of a restricted curve c is a cap of c if s ⊂ H + . It follows that a restricted curve intersecting the baseline in 2n points determines n − 1 caps. Two single-curves p, q ∈ H + are separated by a cap s if p and q lie entirely in two different arc-connected components of H + r s. Two single-curves p, q ∈ H + are separated in a restricted family F if they are separated by some cap of a curve from F. Lemma 3.2. Let F be a restricted family of curves, and let G ⊆ F. If L(c) and R(c) are separated in F for every curve c ∈ G, then χ(G) 6 4. Proof. Let S = {L(c) : c ∈ G} ∪ {R(c) : c ∈ G}. Construct an auxiliary graph G as follows. The S vertices of G are the arc-connected components of S. The edges of G are the pairs of vertices U V for which there is c ∈ G such that L(c) ⊆ U and R(c) ⊆ V or vice versa. Suppose G has a loop at a vertex V . Then there is c ∈ G such that L(c), R(c) ⊂ V . Since L(c) and R(c) are separated in F, there is a cap s of a curve in F such that L(c) and R(c) are separated by s. Therefore, by the Jordan curve theorem, V and s must intersect. This and the S fact that V ⊆ F contradict the assumption that F is restricted. Hence, G has no loops. The graph G is planar. Indeed, each curve c ∈ G such that L(c) ⊆ U and R(c) ⊆ V , witnessing an edge (U, V ) of G, connects U with V in such a way that there are no intersections with other curves in between. This is because any such intersection would contradict the assumption that F is restricted. Thus χ(G) 6 4. Fix a proper 4-coloring of G, and let the color of a curve c ∈ G be defined as the color of the vertex of G containing L(c). This yields a proper 4-coloring of G. Indeed, if two curves c1 , c2 ∈ G, then either R(c1 ) intersects L(c2 ) or L(c1 ) intersects R(c2 ), so the two vertices of G after which c1 and c2 inherit their colors are connected by an edge of G, witnessed by c1 if R(c1 ) intersects L(c2 ) or by c2 if L(c1 ) intersects R(c2 ). Lemma 3.3. Let F be a restricted family of curves, and let c1 , c2 ∈ F. If neither L(c1 ) and R(c1 ) nor L(c2 ) and R(c2 ) are separated in F, then the intervals [l(c1 ), r(c1 )] and [l(c2 ), r(c2 )] are nested or disjoint. Proof. Suppose to the contrary that [l(c1 ), r(c1 )] and [l(c2 ), r(c2 )] are neither nested nor disjoint. Then l(c1 ) < l(c2 ) < r(c1 ) < r(c2 ) or l(c2 ) < l(c1 ) < r(c2 ) < r(c1 ). Assume the former, without loss of generality. Connect the endpoints of c2 by a curve p lying strictly above the baseline and avoiding all other points of c2 . Such a curve exists, otherwise L(c2 ) and R(c2 ) would be separated by a cap of c2 . Let C denote the closed curve obtained by taking the union of p and c2 . For every cap s of c2 , since L(c1 ) and R(c1 ) are not separated by s, the segment [l(c1 ), r(c1 )] contains either both or neither of the endpoints of s. It also contains l(c2 ) but not r(c2 ). It follows that the number of intersections of c2 with the segment [l(c1 ), r(c1 )] is odd. Therefore, by the Jordan curve theorem, l(c1 ) and r(c1 ) lie in two different regions of C. Let x and y be two intersection points of c1 with the baseline such that x is in the same region of C as l(c1 ), y is in the same region of C as r(c1 ), and x, y are consecutive on c1 . Let s1 denote the section of c1 between x and y. It does not intersect c2 , because c2 can only intersect the parts L(c1 ) and R(c1 ) of c1 . Clearly, s1 and C must intersect, which is therefore possible only when s1 and p intersect. This implies that s1 ⊂ H + , because p ⊂ H + . The segment [x, y] intersects C and thus c2 an odd number of times. For every cap s2 of c2 , the segment [x, y] contains either both or neither of the endpoints of s2 , because s1 does not intersect c2 . Therefore, [x, y] contains exactly one of the points l(c2 ), r(c2 ). This means that L(c2 ) and R(c2 ) are separated by s1 , which is a contradiction.
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Lemma 3.4. Suppose there is a function f : N → N such that every restricted family F˜ of curves intersecting the baseline in exactly two points satisfies χ(F˜ ) 6 f (ω(F˜ )). Then every restricted family of curves F satisfies χ(F) 6 f (ω(F)) + 4. Proof. Let F be a restricted family of curves. Let G ⊆ F be the family of those curves c ∈ F for which L(c) and R(c) are separated in F. By Lemma 3.2, χ(G) 6 4. Let H = F r G. By Lemma 3.3, for any c1 , c2 ∈ H, the intervals [l(c1 ), r(c1 )] and [l(c2 ), r(c2 )] are nested or disjoint, and therefore the semicircles in H − with diameters [l(c1 ), r(c1 )] and [l(c2 ), r(c2 )] are disjoint. Transform each curve c ∈ H replacing its part between l(c) and r(c) by a semicircle ˜ with the same with diameter [l(c), r(c)] contained in H − . This way a restricted family H ˜ intersects the baseline in exactly two intersection graph as H is obtained. Every curve in H ˜ 6 f (ω(H)) ˜ = f (ω(H)). We points. Therefore, by the assumption of the lemma, χ(H) = χ(H) conclude that χ(F) 6 χ(G) + χ(H) 6 f (ω(H)) + 4. Lemma 3.5. Suppose there is a (nondecreasing) function f : N → N such that every restricted ˜ 6 f (ω(F)). ˜ family F˜ of curves intersecting the baseline in exactly two points satisfies χ(F) Let f1 : N → N be the function claimed by Theorem 1.3, that is, such that every family H of curves intersecting the baseline in exactly one point satisfies χ(H) 6 f1 (ω(H)). For t > 2, let ( 2 (x) if t is odd, ft−1 ft (x) = 2 (f (x) + 4)ft−1 (x) if t is even. Then, for t > 1, every family F of curves intersecting the baseline in at least one and at most t points satisfies χ(F) 6 ft (ω(F)). Proof. The proof goes by induction on t. There is nothing to prove for t = 1 (the conclusion repeats an assumption), so suppose t > 2. Let F be a family of curves intersecting the baseline in at least one and at most t points. If t is odd, then it follows from Lemma 3.1 that χ(F) 6 2 (ω(F)) = f (ω(F)). Now, suppose t is even. By Lemma 3.1, there is a subfamily G ⊂ F ft−1 t 2 (ω(F)) such that any intersection between two curves c , c ∈ G is with χ(G) > χ(F)/ft−1 1 2 either between R(c1 ) and L(c2 ) or between L(c1 ) and R(c2 ). Since the curves c ∈ G with L(c), R(c) ∈ H + cannot intersect the curves c ∈ G with L(c), R(c) ∈ H − and by symmetry, we can assume without loss of generality that L(c), R(c) ∈ H + for every c ∈ G. This implies that G is restricted and therefore, by Lemma 3.4, χ(G) 6 f (ω(G)) + 4 6 f (ω(F)) + 4. We conclude 2 (ω(F)) 6 (f (ω(G)) + 4)f 2 (ω(F)) as required. that χ(F) 6 χ(G)ft−1 t−1 Theorem 1.6 is now a direct consequence of Lemma 3.5. The exact same argument as in Lemmas 3.1–3.5 but for triangle-free families of simple curves intersecting a fixed line in at least one and at most t points gives the following reduction. Lemma 3.6. Suppose there is a constant α such that every triangle-free simple restricted family F˜ of curves intersecting the baseline in exactly two points satisfies χ(F˜ ) 6 α. Let β1 be the constant claimed by Theorem 1.2, that is, such that every triangle-free simple family H of curves intersecting the baseline in exactly one point satisfies χ(H) 6 β1 . For t > 2, let ( 2 if t is odd, βt−1 βt (x) = 2 (α + 4)βt−1 if t is even. Then, for t > 1, every triangle-free simple restricted family F of curves intersecting the baseline in at least one and at most t points satisfies χ(F) 6 βt .
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s2 s 1 p1
p2 p3
p4
Figure 1. A family {p1 , p2 , p3 , p4 } externally supported by s1 and s2 . 4. Curves intersecting a fixed line exactly twice This section is devoted to the proof of Theorem 1.5. 4.1. Preliminaries. The exterior of a family of curves F, denoted by ext(F), is the unique S unbounded arc-connected component of R2 r G. A family G ⊂ F is externally supported in F if for every curve p ∈ G there is a curve s ∈ F that intersects p and ext(G). Such an s is called the support of p when the family G is implicit. The following lemma is a straightforward application of the ideas expressed by Gy´arf´as [6, 7], which were subsequently used in [16, 17, 22, 14, 21]. Lemma 4.1. Every family of curves F with χ(F) > 2 has a subfamily G that is externally supported in F and satisfies χ(G) > χ(F)/2. Proof. We can assume without loss of generality that the intersection graph of F is connected, otherwise we can restrict ourselves to the connected component with maximum chromatic number. Let c0 be an arbitrary curve in F that intersects the boundary of ext(F) at a point that does not belong to any other curve in F. For i > 0, let Fi denote the family of curves in F that are at distance i from c0 in the intersection graph of F. It follows that F0 = {c0 }, S S S F = i>0 Fi = {c0 } ∪ F , and each curve in Fi is disjoint from each i>0 F2i+1 ∪ i>0 S 2i+2 S curve in Fj whenever |i − j| > 1. Thus χ( i>0 F2i+1 ) > χ(F)/2 or χ( i>0 F2i+2 ) > χ(F)/2, so there is d > 1 with χ(Fd ) > χ(F)/2. We claim that Fd is externally supported in F. Fix cd ∈ Fd , and let c0 . . . cd be a shortest path from c0 to cd in the intersection graph of F. Since c0 hits the boundary of ext(F), it S intersects ext(Fd ). Moreover, c0 , . . . , cd−2 are disjoint from Fd , as otherwise there would be a curve in Fd at distance less than d from c0 . Therefore, all c0 , . . . , cd−2 are entirely contained in ext(Fd ). This implies that cd−1 intersects ext(Fd ), so it is a support of cd . A closed curve is a homeomorphic image of a circle. By the Jordan curve theorem, for every closed curve C, the set R2 r C splits into two arc-connected components, one bounded, called the interior of C and denoted by int(C), and one unbounded, called the exterior of C and denoted by ext(C). The following is a special case of the main result proved in [18]. Lemma 4.2 (McGuinness [18]). There are constants B and K with the following property. Let F be a family of curves such that any two curves in F intersect in at most one point and the intersection graph of F is triangle-free. Then there is a closed curve C constained in the union
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ALEXANDRE ROK AND BARTOSZ WALCZAK
of at most 5 curves from F such that if G ⊂ F is the family of curves from G contained in int(C), then χ(G) > χ(F)/K. 4.2. General setting. Lemma 3.6 reduces Theorem 1.5 to triangle-free simple restricted families of curves F with exactly two intersection points with the baseline. We fix this setting for the remainder of Section 4. Namely, we let F be a family of curves with the following properties: (1) Each curve in F intersects the baseline in exactly two points, both of which are proper intersections. (2) Both endpoints of each curve in F lie above the baseline. (3) Any intersection between two curves in F is an intersection between their single-curves of opposite types, that is, an intersection of a right single-curve with a left single-curve. (4) Any two curves in F intersect in at most one point. (5) The intersection graph of F is triangle-free. We remark that the only reason for the conditions 4 above is that it is required by Lemma 4.2. Proving an analogue of Lemma 4.2 for more general classes of curves (with any constant in place of 5) will automatically lead to a generalization of our Theorem 1.5. When s is a single-curve arising from a curve in F, then we write C(s) to denote the curve from F that gives rise to s. If S is a family of single-curves arising from curves in F, then we define C(S) = {C(s) : s ∈ S}. 4.3. Getting surrounded. If p1 < · · · < ps are the intersection points of a curve with the baseline, then we call the points pi and pi+1 consecutive for 1 6 i 6 s − 1. Note that this concept of being consecutive differs from the one used in Section 3. Lemma 4.3. Let C be a closed curve, and let a and b be two consecutive intersection points of C with the baseline such that (a, b) ⊂ int(C). If L, R ⊂ F are families of curves contained in int(C) such that only the left basepoint and only the right basepoint is in (a, b), respectively, then χ(L) 6 1 and χ(R) 6 1. Proof. We prove only that χ(L) 6 1, since the other case is analogous. Let c1 and c2 be two intersecting curves of L. If b′ is the next intersection point of C with the baseline after b, then for both c1 and c2 the interval between their basepoints must contain (b, b′ ). Without loss of generality L(c1 ) ≺ L(c2 ). Since no intersection between c1 and c2 can occur below the baseline, it follows that R(c2 ) ≺ R(c1 ). If L(c1 ) intersects R(c2 ) and p denotes their first intersection point as going along L(c1 ) from the basepoint, then the subcurve of L(c1 ) from the basepoint to p, the subcurve of c2 from p to the right basepoint of c2 , the interval [b(L(c1 )), b(L(c2 ))] and the subcurve of c2 connecting b(L(c2 )) and b(R(c2 )) form a closed curve contained in int(C) that contains the segment (b, b′ ) in its interior. This implies (b, b′ ) ⊂ int(C), contradicting the assumption that (a, b) ⊂ int(C). The case that R(c1 ) intersects L(c2 ) is analogous. Lemma 4.4. Let C be a closed curve contained in the union of at most 5 curves from F. If G ⊂ F and every curve in G is contained in int(C), then there are two consecutive intersection points a, b of C with the baseline such that χ(G(a, b)) > χ(G)/5 − 1. Proof. Let p1 < · · · < p2n be the intersection points of C with the baseline. For 1 6 k 6 n, let Gk denote the curves from G whose left basepoint is in (p2k+1 , p2k+2 ). Since C is contained in the union of at most 5 curves intersecting the baseline twice, we have n 6 5. Therefore, for some k ∈ {1, . . . , n}, we have χ(Gk ) > χ(G)/5. By Lemma 4.3, the curves from Gk whose right basepoint is not in (p2k+1 , p2k+2 ) have chromatic number at most 1. The remaining curves form the family G(p2k+1 , p2k+2 ), which has chromatic number at least χ(G)/5 − 1.
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p1
p2
p3
9
p4
Figure 2. A skeleton of type 1 supporting some family {p1 , p2 , p3 }, while p4 is not supported. 4.4. Rising of skeletons. A skeleton is a pair (C, S), where C is a closed curve and S is a family of single-curves of the same type (all left or all right) arising from curves in F such that the basepoints of the single-curves in S belong to int(C). Since all members of S in a skeleton (C, S) are of the same type, they are pairwise disjoint. The initial part of a single-curve s ∈ S with respect to a skeleton (C, S) is the part of s between b(s) and the first intersection point of s with C. The initial part of s is denoted by s′ when the skeleton is clear from the context. A family G ⊂ F is supported by the skeleton (C, S) if every curve in G is contained in int(C) and intersects s′ for some s ∈ S. See Figure 2. Lemma 4.5. Let gα (x) = 2K(max{(20x + 5)K, B} + 2α), where K and B are the constants from Lemma 4.2. Let F˜ ⊂ F. If χ(F˜ ) > gα (x), then at least one of the following holds: (1) There is a subfamily G ⊂ F˜ with χ(G) > α supported by a skeleton (C, S) such that C is ˜ contained in the union of at most 5 curves from F˜ and S ⊂ F. ˜ (2) There is a subfamily P ⊂ F with χ(G) > x and a family of single-curves Y of common type arising from some curves from F˜ such that each p ∈ P intersects some s ∈ Y and P ≺ Y or Y ≺ P. Proof. Suppose the condition 1 does not hold. We apply Lemma 4.1 in order to get an externally ˜ supported subfamilly G ⊂ F˜ with χ(G) > χ(F)/2 > K(max{(20x + 5)K, B} + 2α). Next, we apply Lemma 4.2 to G to get a closed curve C contained in the union of at most 5 curves from ˜ > χ(G)/K > G such that the family G˜ ⊂ G of curves from G contained in int(C) satisfies χ(G) max{(20x + 5)K, B} + 2α. Let S denote the set of supports of G, that is, the set of curves from F˜ intersecting some curve from G and ext(G). Let S1 ⊂ L(S) and S2 ⊂ R(S) be the sets of those curves of L(S) and R(S), respectively, that are based in int(C). Every curve in S1 or S2 S intersects ext(C), because it intersects ext(G) and C ⊂ G. After removing from G˜ all curves supported by at least one of the skeletons (C, S1 ), (C, S2 ), we are left with a family H ⊂ G˜
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˜ − 2α > max{(20x + 5)K, B}. Apply Lemma 4.2 again to H to get a such that χ(H) > χ(G) S ˜ ˜ of curves from H contained in int(C) ˜ satisfies closed curve C ⊂ H such that the family H ˜ > χ(H)/K > 20x + 5. By Lemma 4.4, there are two consective intersection points a and χ(H) ˜ and χ(H(a, ˜ b)) > χ(H)/5 ˜ b of C˜ with the baseline such that (a, b) ⊂ int(C) − 1 > 4x. ˜ b). If one basepoint of s lies in int(C) ˜ and Suppose s ∈ S is a support of a curve c ∈ H(a, ˜ ˜ the other in ext(C), then s must intersect C below the baseline, which is impossible in our ˜ setting. Suppose both basepoints of s lie in int(C). Then, since no intersection can occur ˜ However, below the baseline, the whole subcurve of s connecting l(s) and r(s) lies in int(C). ˜ as well. This implies, by the Jordan by assumption, s must intersect ext(C) and hence ext(C) curve theorem, that at least one of the single-curves of s intersects C˜ and then C as going along S the single-curve from the basepoint. In particular s ∈ Sk for some k ∈ {1, 2}. Since C˜ ⊂ H, the intersection point of that single-curve of s with C˜ belongs to a curve p ∈ H, as well as it
belongs to the initial part s′ of s with respect to the skeleton (C, Sk ). Therefore, p is supported by (C, Sk ), which contradicts the definition of H. This shows that every support of a curve in ˜ b) has both bases in ext(C). ˜ H(a, ˜ ˜ b) The family H(a, b) is the union of B1 and B2 , where B1 is the set of all curves c ∈ H(a, having a support s ∈ S such that L(s) intersects c and symmetrically for B2 with R(s). Clearly, ˜ b))/2 > 2x. Let S˜k be the family of single-curves for some k ∈ {1, 2}, we have χ(Bk ) > χ(H(a, of type k arising from the corresponding supports. Now Bk is the union of two sets A1 and A2 , where A1 is the set of all curves of Bk that are intersected by some single-curve of S˜k based to the left of Bk and symmetrically for A2 . For some j ∈ {1, 2}, we have χ(Aj ) > χ(Bk )/2 > x. Let P = Aj and Y be the subset of S˜k corresponding to Aj . It follows that the condition 2 is satisfied for P and Y. For a function h : R → R, we let h(m) denote the m-fold composition of h. (3) Lemma 4.6. Let f (α) = gα (4), where g is the function from Lemma 4.5. Let F˜ ⊂ F. If χ(F˜ ) > f (α), then there is a subfamily G ⊂ F˜ with χ(G) > α supported by a skeleton (C, S) such that C is contained in the union of at most 5 curves from F˜ and S ⊂ F˜ .
˜ For 1 6 i 6 3, let Fi Proof. Suppose that such a subfamily G does not exist. Let F0 = F. and Yi be the families obtained by applying the conclusion 2 of Lemma 4.5 to Fi−1 . That is, (3−i) Fi ⊂ Fi−1 , χ(Fi ) > gα (4), Yi is a family of single-curves of common type arising from some curves in Fi−1 , each c ∈ Fi intersects some s ∈ Yi , and Fi ≺ Yi or Yi ≺ Fi . Two of the Yi s, say Yi1 and Yi2 where 1 6 i1 < i2 6 3, must contain single-curves of the same type. We restrict ourselves only to the case that Yi2 ≺ Fi2 , as the opposite one is symmetric. Let X be the family of single-curves of the curves in Fi2 with the type opposite to that of Yi2 , and let Y = Yi2 . Process the single-curves in X in the order of their bases from right to left, adding them to a (3−i ) family H until χ(C(H)) = 2. Let D = X r H, so that χ(C(D)) > gα 2 (4) − 2 > 2. For c ∈ H, let p(c) be the first intersection point of c with a single-curve of Y encountered as going from b(c) along c, let s(c) be the single-curve of Y that intersects c at p(c), and let A(c) be the region enclosed in the closed curve formed by the subcurve of c (denoted by c˜) from b(c) to p(c), the subcurve of s(c) (denoted by s˜(c)) from b(s(c)) to p(c), and the part of the baseline between b(c) and b(s(c)). Let c be the single-curve in H with leftmost base. We claim that A(c) ⊂ A(c′ ) for any other single-curve in H. Indeed, it this is not true, then the boundary curve of A(c′ ) must hit A(c) at some point q when following counterclockwise starting from b(c′ ). Since the pairs of curves c, c′ and s(c), s(c′ ) have common types, they cannot intersect, so q is the intersection of c˜ and s(c′ ) or c˜′ and s(c). It follows that q = p(c) in the former and q = p(c′ ) in the latter case. In both cases s(c′ ) = s(c), so the boundary curve of A(c′ ) cannot enter the interior of A(c).
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˜ such that After removing from D all single-curves intersecting s˜(c), we obtain a family D S˜ ˜ D ⊂ A(c) and χ(C(D)) > χ(C(D)) − 1 > 1. Indeed, since the family F is triangle-free, the set of curves corresponding to the single-curves removed from D must be independent. Now, let ˜ and take any single-curve si from Yi intersecting v. For each single-curve c′ ∈ H, since H v∈D 1 1 and Y are single-curves of curves in Fi1 , si1 ∈ Yi1 , Fi1 ≺ Yi1 or Yi1 ≺ Fi1 , and v ⊂ A(c) ⊂ A(c′ ), the single-curve si1 must intersect c˜′ or s˜(c′ ). It cannot intersect s˜(c′ ), because s(c′ ) and si1 have the same type. Hence si1 intersects all single-curves in H. This yields χ(C(H)) 6 1, which is a contradiction. 4.5. Skeletons rule the world. To complete the proof of Theorem 1.5, we make use of families supported by skeletons to prove that if the chromatic number of F is too large, then it contains the intersection graph of F constains a triangle. If C is a closed curve intersecting the baseline, then an upper section of C is an arc-connected component of H + r C. Lemma 4.7. Let f˜(x) = f (5x), where f is the function defined in Lemma 4.6. Then χ(F) < f˜(11) (511 K(max{20K, B} + 11)), where K and B are the constants from Lemma 4.2. Proof. Suppose to the contrary that χ(F) > f˜(11) (511 K(max{20K, B} + 11)). Let F0 = F. For 1 6 i 6 11, proceed as follows. Apply Lemma 4.6 to Fi−1 to get a subfamily F˜i ⊂ Fi−1 with χ(F˜i ) > 5f˜(11−i) (511 K(max{20K, B} + 11)) supported by a skeleton (Ci , S˜i ), where Ci is a closed curve contained in the union of at most 5 curves from Fi−1 . Let p1 < · · · < p2n be the intersection points of Ci with the baseline. For 1 6 j 6 n, let Sij = S˜i (p2j−1 , p2j ), and let Fij consist of the curves from F˜i that are supported by the skeleton (Ci , Sij ). Thus S S S˜i = nj=1 Sij and F˜i = nj=1 Fij . Since n 6 5, there is j ∈ {1, . . . , n} such that χ(Fij ) > χ(F˜i )/5 > f˜(11−i) (511 K(max{20K, B} + 11)). Let Fi = F j and Si = S j . i
i
Apply Lemma 4.2 to F11 to get a closed curve C contained in the union of r 6 5 curves c1 , . . . , cr ∈ F11 and the subfamily G0 ⊂ F11 of curves contained in int(C) such that χ(G0 ) > χ(F11 )/K > 511 (max{20K, B} + 14). The part of C above the baseline is contained in the union of 2r 6 10 single-curves h1 , . . . , h2r (r of either type) arising from c1 , . . . , cr . Then, for 1 6 i 6 11, proceed as follows. For 1 6 j 6 2r, let Uij be the family of supports s ∈ Si such that the initial part s′ of s with respect to the skeleton (Ci , Si ) intersects hj , and let Gij be the set S S2r j j of curves from Gi−1 that have a support in Uij . Thus Si = 2r j=1 Ui and Gi−1 = j=1 Gi . Since r 6 5 and the supports in Si can only intersect single-curves of h1 , . . . , h2r of the opposite type, at most 5 of the families Uij and thus Gij are nonempty. Therefore, there is j ∈ {1, . . . , 2r} such that χ(Gij ) > χ(Gi−1 )/5 > 511−i (max{20K, B} + 11). Let Gi = Gij and Ui = Uij . Since 2r 6 10, there are two indices i1 and i2 with 1 6 i1 < i2 6 11 such that Ui1 = Uij1 and Ui2 = Uij2 for a common index j ∈ {1, . . . , 2r}. In particular, Ui1 and Ui2 are of the same type, which is opposite to the type of hj . Let h = hj . Let s1 ≺ · · · ≺ sm be the single-curves in Ui1 . Let p1 < · · · < p2n be the intersection points of Ci1 with the baseline. Let U be the upper section of int(Ci1 ) containing (p2j−1 , p2j ). Recall that there is j ∈ {1, . . . , n} such that b(s1 ), . . . , b(sm ) ∈ (p2j−1 , p2j ). The initial parts s′1 , . . . , s′m of s1 , . . . , sm with respect to the skeleton (Ci1 , Ui1 ) are contained in U (except for the endpoints of s′1 , . . . , s′m other than their basepoints). Since s′1 , . . . , s′m are pairwise disjoint, the orders of the endpoints of s′1 , . . . , s′m along the two parts of the boundary of U between p2j−1 and p2j are the same. Therefore, the set U r (s′1 ∪ · · · ∪ s′m ) splits into m + 1 arc-connected components U0 , . . . , Um such that • the boundary of U0 is formed by s′1 , [p2j−1 , b(s1 )], and the clockwise part of the boundary of U from p2j−1 to the other endpoint of s′1 ;
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ALEXANDRE ROK AND BARTOSZ WALCZAK
• for 1 6 k 6 m−1, the boundary of Uk is formed by s′k , s′k+1 , [b(sk ), b(sk+1 )], and the clockwise part of the boundary of U from the other endpoint of s′k to the other endpoint of s′k+1 ; • the boundary of Um is formed by s′m , [b(sm ), p2j ], and the clockwise part of the boundary of U from the other endpoint of s′m to p2j . Let X be the family of single-curves of the type opposite to s1 , . . . , sm arising from the curves in G11 . Thus every single-curve in X intersects s′k for some k ∈ {1, . . . , m}. The above implies that • (p2j−1 , b(s1 )) ⊂ U0 , so every single-curve in X (p2j−1 , b(s1 )) intersects s′1 ; • for every i ∈ {1, . . . , n} r {j}, if X (p2i−1 , p2i ) 6= ∅, then (p2i−1 , p2i ) ⊂ U ; consequently, there is ki ∈ {0, . . . , m} such that (p2i−1 , p2i ) ⊂ Uki , so every single-curve in X (p2i−1 , p2i ) intersects s′ki or s′ki +1 ; • (b(sm ), p2j ) ⊂ Um , so every single-curve in X (b(sm ), p2j ) intersects s′m . ˜ be the curves from G11 that do not intersect any of s′ , s′m , and s′ , s′ Let H 1 ki +1 for i ∈ ki ˜ {1, . . . , n} r {j}. Since n 6 5, it follows that χ(H) > χ(G11 ) − 10 > max{20K, B} + 1 Now, let ˜ is formed by s′ , s′m , [b(s1 ), b(sm )], ˜ = s′ ∪ U1 ∪ s′ ∪ · · · ∪ Um−1 ∪ s′m , so that the boundary of U U 1
2
1
and the clockwise part of the boundary of U from the other endpoint of s′1 to the other endpoint ˜ are of s′m . The single-curves of the type opposite to s1 , . . . , sm arising from the curves in H ˜ ˜ ˜ contained in U . The other single-curves of the curves in H are either contained in U or disjoint ˜ . Those curves from H ˜ whose other single-curves are contained in U ˜ form the family from U ˜ ˜ have chromatic H = H(b(s1 ), b(sm )), while those whose other single-curves are disjoint from U ˜ − 1 > max{20K, B}. number at most 1. Hence χ(H) > χ(H) Apply Lemma 4.2 to H to get a closed curve C˜ contained in the union of at most 5 curves ˜ such that χ(D) > χ(H)/K > 20. from H and the subfamily D ⊂ H of curves contained in int(C) ˜ Let q1 < · · · < q2n be the intersection points of C with the baseline. By Lemma 4.4, there is j ∈ {1, . . . , n} such that χ(D(q2j−1 , q2j )) > χ(D)/5 − 1 > 3. By an argument analogous to the one above, it can be proved that the curves from D(q2j−1 , q2j ) supported by (Ci2 , Ui2 rUi2 (q2j−1 , q2j )) intersect two specific supports from Ui2 r Ui2 (q2j−1 , q2j ) and hence have chromatic number at most 2. Therefore, since χ(D(q2j−1 , q2j )) > 3, at least one curve from D(q2j−1 , q2j ) must be supported by (Ci2 , Ui2 (q2j−1 , q2j )), so Ui2 (q2j−1 , q2j ) 6= ∅. Let x ∈ Ui2 (q2j−1 , q2j ), and let x ˜ be the opposite single-curve of C(x). It follows from ˜ that b(˜ ˜ otherwise C(x) would intersect C˜ below the b(x) ∈ (q2j−1 , q2j ) ⊂ int(C) x) ∈ int(C), baseline, which is impossible. This implies b(˜ x) ∈ (b(s1 ), b(sm )), because all intersection points of C˜ with the baseline lie in (b(s1 ), b(sm )). Since C(x) ∈ Fi1 , there is s ∈ Si1 that supports C(x) in the skeleton (Ci1 , Si1 ). Since Si1 and x have the same type, s intersects x ˜. Recall that each single-curve in Ui1 or Ui2 intersects h. In particular, so do s1 , sm and x. Let H be the closed curve formed by the segment [b(s1 ), b(sm )] of the baseline, the part of s1 from b(s1 ) to the first intersection point with h, the part of sm from b(sm ) to the first intersection point with h, and the part of h between the two intersection points. It follows that ˜ have the same type, at least one of the following int(H) ⊂ int(C) ⊂ int(Ci1 ). Since h and x holds: (1) x ˜ intersects s1 , (2) x ˜ intersects sm , or (3) x ˜ intersects s at a point in int(H). In case (1) or (2), we can assume without loss of generality that s = s1 or s = sm , respectively. In case (3), s must go out of int(H) in order to intersect Ci1 , which is only possible by crossing h, because s, s1 and sm have the same type. In each of the three cases, C(h), C(s) and C(x) form a triangle in the intersection graph, which is a contradiction. Proof of Theorem 1.5. Lemma 4.7 shows that triangle-free simple restricted family of curves each intersecting the baseline in exactly two points have bounded chromatic number. Therefore, the assumption of Lemma 3.6 is satisfied, and Theorem 1.5 directly follows.
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(Alexandre Rok) Department of Mathematics, Ben-Gurion University of the Negev, Be’er Sheva 84105, Israel E-mail address: [email protected] (Bartosz Walczak) Theoretical Computer Science Department, Faculty of Mathematics and Computer Science, Jagiellonian University, Krak´ ow, Poland E-mail address: [email protected]
