Coloring curves that cross a fixed line - Semantic Scholar
COLORING CURVES THAT CROSS A FIXED LINE ALEXANDRE ROK AND BARTOSZ WALCZAK Abstract. Let F be a family of curves in the plane with the following properties:
arXiv:1512.06112v2 [math.CO] 9 Mar 2016
(1) each member of F crosses a fixed straight line L at least once and at most t times, (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 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) 6 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 is not χ-bounded. 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, proposed by McGuinness [16, 17], is that each curve must cross a fixed line in exactly one point. Theorem 1.2 (McGuinness [17]). Triangle-free simple families of curves each crossing 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 crossing a fixed line in exactly one point is χ-bounded. Alexandre Rok was partially supported by the Israel Science Foundation under grant 1136/12. Bartosz Walczak was partially supported by the National Science Center of Poland under grant 2015/17/D/ST1/00585. 1
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ALEXANDRE ROK AND BARTOSZ WALCZAK
Do the statements from Theorems 1.2 and 1.3 remain valid if we allow the curves to cross the fixed line in more than one point? In view of Theorem 1.1, we cannot hope for any similar result for families of curves crossing 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, the class of intersection graphs of families of curves crossing a fixed line in at least one and at most t points is χ-bounded. Results. We present two results which offer the first progress towards resolving Conjecture 1.4. Theorem 1.5. For every t, triangle-free simple families of curves crossing 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 crossing 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 crossing 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. 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 axis-parallel 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 circle graphs—intersection graphs of families of chords of a fixed circle. Gy´arf´as [6, 7] proved that circle 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 known construction of circle graphs with large chromatic number forces Ω(ω log ω) colors [9]. McGuinness [16, 17] proposed to investigate the problem allowing much more general geometric shapes but restricting the way how they can be positioned in the plane. First, he proved that the class of intersection graphs of L-shapes crossing a fixed horizontal line is χ-bounded [16]. Then, he also showed that triangle-free simple families of curves crossing 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 crossing a fixed vertical line give rise to a χ-bounded class of graphs, and by Laso´ n et al. [14], who showed
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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 crossing 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 intersection graphs of segments or, more generally, x-monotone curves, this can be improved to Oω (log n) using the above-mentioned result of Suk [22] or Rok and Walczak [21] and 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 the construction 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 admit a curvilinear plane drawing with no k pairwise crossing edges. A well-known conjecture asserts that n-vertex k-quasi-planar graphs 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 one point and c 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 as the baseline. It is the line that curves from the families that we consider are required to cross. For any considered family of curves F, we assume that every crossing between a curve c ∈ F and 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 cross the baseline at a common point. The closed halfplanes above and below the baseline are denoted by H + and H − , respectively. The left-to-right order of the points on the baseline is denoted by 2. Suppose there is a nondecreasing function ft−1 : N → N such that every family G of curves crossing the baseline at least once and at most t − 1 times satisfies χ(G) 6 ft−1 (ω(G)). Let F be a family of curves crossing the baseline at least once and at 2 (ω(F)) such that any most t times. 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 1-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. Construct two families F1 and F2 of curves crossing the baseline at least once and at most t − 1 times, as follows: (a) for every curve c ∈ F crossing the baseline exactly t times, put c L(c) to F1 and c R(c) to F2 , (b) put every curve from F crossing the baseline less than t times to both F1 and F2 . This yields χ(Fk ) 6 ft−1 (ω(Fk )) 6 ft−1 (ω(F)) for k ∈ {1, 2}. For c ∈ F and k ∈ {1, 2}, let φk (c) be the color of the curve obtained from c in a 2 (ω(F))-coloring proper ft−1 (ω(F))-coloring of Fk . This yields a (not necessarily proper) ft−1 2 (ω(F)) and the restriction of (φ , φ ) (φ1 , φ2 ) of F. There is G ⊆ F such that χ(G) > χ(F)/ft−1 1 2 to G is constant. Since any two curves c1 , c2 ∈ F with (φ1 (c1 ), φ2 (c1 )) = (φ1 (c2 ), φ2 (c2 )) can intersect only when their opposite 1-curves intersect, G is as required in the lemma. A restricted curve is a curve c crossing the baseline such that L(c), R(c) ⊆ H + . It follows that every restricted curve crosses the baseline an even number of times. 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 ). A cap of a restricted curve c is a part of c between two consecutive basepoints that lies in H + . Every curve in a restricted family F is disjoint from all caps of the other curves in F. Two 1-curves x, y ∈ H + are separated in a restricted family F if they lie entirely in two different arc-connected components of H + r C, where C is the union of all caps of curves in 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, they lie in different arc-connected components of F r C, where C is the union of all caps of curves in F. Therefore, V and C intersect, which implies that some 1-curve of a curve from F forming the vertex V intersects some cap of a curve from F. This contradicts the assumption that F is restricted. Therefore, G has no loops. Each curve c ∈ G such that L(c) ⊆ U and R(c) ⊆ V , witnessing an edge U V of G, connects U and V in such a way that there are no intersections with other curves in between. This is
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because any such intersection would contradict the assumption that F is restricted. Therefore, G is planar, and thus χ(G) 6 4. Let φ be a proper 4-coloring of G. Color the curves in G so that the color φ(V ) of every vertex V of G is applied to every curve c ∈ G such that L(c) ⊆ V . For any c1 , c2 ∈ G, if R(c1 ) and L(c2 ) intersect, then they are part of the same vertex of G whose color is different from the color of the vertex of G containing L(c1 ). Therefore, the 4-coloring of G is proper. 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 [l(c1 ), r(c1 )] and [l(c2 ), r(c2 )] are neither nested nor disjoint, that is, l(c1 ) < l(c2 ) < r(c1 ) < r(c2 ) or l(c2 ) < l(c1 ) < r(c2 ) < r(c1 ). For k ∈ {1, 2}, since L(ck ) and R(ck ) are not separated in F, the points l(ck ) and r(ck ) can be connected by a curve cˆk ⊆ H + that avoids all caps of curves from F. We can assume without loss of generality that cˆ1 and cˆ2 cross in exactly one point p. For k ∈ {1, 2}, let γk be the closed curve obtained as the union of cˆk and the part of ck between l(ck ) and r(ck ). It follows that p is the unique crossing point of γ1 and γ2 , which contradicts the Jordan curve theorem. Lemma 3.4. If every restricted family F˜ of curves crossing the baseline exactly twice satisfies ˜ 6 f (ω(F)), ˜ χ(F) where f : N → N is a nondecreasing function, 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 − , thus obtaining a restricted family H ˜ crosses the baseline exactly twice. Therefore, intersection graph as H. Every curve in H ˜ 6 f (ω(H)) ˜ = f (ω(H)). We conclude that by the assumption of the lemma, χ(H) = χ(H) χ(F) 6 χ(G) + χ(H) 6 f (ω(H)) + 4 6 f (ω(F)) + 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 crossing the baseline exactly twice satisfies χ(F) Let f1 : N → N be the function claimed by Theorem 1.3, that is, such that every family H of curves crossing 2 (x). the baseline exactly once satisfies χ(H) 6 f1 (ω(H)). For t > 2, let ft (x) = (f (x) + 4)ft−1 Then, for t > 1, every family F of curves crossing the baseline at least once and at most t times satisfies χ(F) 6 ft (ω(F)). Proof. The proof goes by induction on t. There is nothing to prove for t = 1, so suppose 2 (ω(F)) such that any intersection t > 2. By Lemma 3.1, there is G ⊆ F with χ(G) > χ(F)/ft−1 between two curves c1 , c2 ∈ G is either between R(c1 ) and L(c2 ) or between L(c1 ) and R(c2 ). If t is odd, then no such intersection can occur, and thus χ(G) 6 1. Now, suppose t is even. Let G + = {c ∈ G : L(c), R(c) ⊆ H + } and G − = {c ∈ G : L(c), R(c) ⊆ H − }. Then G = G + ∪ G − and χ(G) = max{χ(G + ), χ(G − )}. Since G + is a restricted family, Lemma 3.4 yields χ(G + ) 6 f (ω(G + )) + 4 6 f (ω(F)) + 4. By symmetry, we also have χ(G − ) 6 f (ω(F)) + 4. We 2 (ω(F)) 6 (f (ω(F)) + 4)f 2 (ω(F)). conclude 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 crossing a fixed line at least once and at most t times gives the following reduction.
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s2 s1 p1
p2 p3
p4
Figure 1. A family {p1 , p2 , p3 , p4 } externally supported by s1 and s2 Lemma 3.6. Suppose there is an integer constant α such that every triangle-free simple restricted ˜ 6 α. Let β1 be the constant family F˜ of curves crossing the baseline exactly twice satisfies χ(F) claimed by Theorem 1.2, that is, such that every triangle-free simple family H of curves crossing 2 . Then, for t > 1, the baseline exactly once satisfies χ(H) 6 β1 . For t > 2, let βt = (α + 4)βt−1 every triangle-free simple restricted family F of curves crossing the baseline at least once and at most t times satisfies χ(F) 6 βt . 4. Curves crossing the baseline 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 c ∈ G there is a curve s ∈ F that intersects c and ext(G). See Figure 1. Such a curve s is called the support of c 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 .
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A closed curve is a homeomorphic image of a circle. By the Jordan curve theorem, for every closed curve γ, the set R2 r γ splits into two arc-connected components, one bounded, called the interior of γ and denoted by int(γ), and one unbounded, called the exterior of γ and denoted by ext(γ). For a family of curves F and a closed curve γ, we let F(γ) denote the family of curves from F that lie entirely in int(γ). The following is a special case of the main result proved in [18]. Lemma 4.2 (McGuinness [18]). There are integer 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, the intersection graph of F is triangle-free, and χ(F) > B. Then there is a closed curve γ contained in the union of at most 5 curves from F such that χ(F(γ)) > 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. This is the common setting for the rest of this section—in particular, for Lemmas 4.3–4.6. Namely, we let F be a family of curves with the following properties: (1) Each curve in F crosses the baseline in exactly two points, both of which are proper crossings. (2) Both endpoints of each curve in F lie above the baseline. (3) Any intersection between two curves in F is between their 1-curves of opposite types, that is, between a right 1-curve and a left 1-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 condition (4) is that it is required by Lemma 4.2. Proving an analogue of Lemma 4.2 for more general classes of curves will automatically lead to a generalization of our Theorem 1.5. 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 that used in Section 3. Lemma 4.3. Let γ be a closed curve contained in the union of at most 5 curves from F. If G ⊆ F(γ), then there are two consecutive intersection points a, b of γ with the baseline such that χ(G(a, b)) > χ(G)/5 − 1. Proof. Let p1 < · · · < p2n be the intersection points of γ with the baseline. For 1 6 k 6 n, let Gk = {c ∈ G : l(c) ∈ (p2k−1 , p2k )}. Since γ is contained in the union of at most 5 curves each crossing the baseline twice, we have n 6 5. Therefore, for some k ∈ {1, . . . , n}, we have χ(Gk ) > χ(G)/5. It is enough to show that χ(Gk r G(p2k−1 , p2k )) 6 1. To this end, suppose c1 , c2 ∈ Gk r G(p2k−1 , p2k ) and L(c1 ) ∩ R(c2 ) = {p}. Then k < n and r(c2 ) ∈ (p2k+1 , p2n ). The subcurve of L(c1 ) from l(c1 ) to p, the subcurve of R(c2 ) from p to l(c2 ), and the interval between l(c1 ) and l(c2 ) form a closed curve contained in int(γ) whose interior contains the interval (p2k , p2k+1 ). This is a contradiction. 4.4. Rising of skeletons. A skeleton is a pair (γ, S), where γ is a closed curve, S ⊆ L(F) or S ⊆ R(F), every 1-curve in S intersects γ, and the basepoint of every 1-curve in S lies in int(γ). Since all members of S in a skeleton (γ, S) are of the same type, they are pairwise disjoint. The initial part of a 1-curve s ∈ S with respect to a skeleton (γ, S) is the part of s between b(s) and the first intersection point of s with γ. The initial part of s is denoted by s0 when the skeleton is clear from the context. A family G ⊆ F(γ) is supported by the skeleton (γ, S) if every curve c ∈ G intersects s0 for some 1-curve s ∈ S, which is called the support of c in (γ, S). See Figure 2.
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p1
p2
p3
p4
Figure 2. A skeleton supporting a family {p1 , p2 , p3 } (p4 is not supported) Lemma 4.4. Let gα (β) = 2K(max{(20β + 5)K, B} + 2α), where K and B are the constants from Lemma 4.2. If χ(F) > gα (β), then one of the following holds: (1) There is a family G ⊆ F with χ(G) > α supported by a skeleton (γ, S) such that γ is contained in the union of at most 5 curves from F and S ⊆ L(F) or S ⊆ R(F). (2) There is a family G ⊆ F with χ(G) > β and a family S ⊆ L(F) or S ⊆ R(F) such that each c ∈ G intersects some s ∈ S and G ≺ S or S ≺ G. Proof. Suppose (1) does not hold. Apply Lemma 4.1 to get an externally supported subfamily G ⊆ F with χ(G) > χ(F)/2 > K(max{(20β + 5)K, B} + 2α). Next, apply Lemma 4.2 to G to get a closed curve γ contained in the union of at most 5 curves from G such that χ(G(γ)) > χ(G)/K > max{(20β + 5)K, B} + 2α. Let U denote the set of supports of G, that is, the set of curves from F intersecting ext(G) and some curve from G. Let SL = {s ∈ L(U) : b(s) ∈ int(γ)} and SR = {s ∈ R(U) : b(s) ∈ int(γ)}. Every 1-curve in SL and SR S intersects ext(γ), because it intersects ext(G) and γ ⊆ G. Removing all curves supported by at least one of the skeletons (γ, SL ), (γ, SR ) from G(γ) yields a family H ⊆ G(γ) such that χ(H) > χ(G(γ)) − 2α > max{(20β + 5)K, B}. Apply Lemma 4.2 again to H to get a closed S curve ξ ⊆ H such that χ(H(ξ)) > χ(H)/K > 20β + 5. Consider a support u ∈ U of a curve from H(ξ). Since u cannot intersect ξ below the baseline, the basepoints of u lie both in int(ξ) or both in ext(ξ). Suppose l(u), r(u) ∈ int(ξ). Then the whole subcurve of u between l(u) and r(u) lies in int(ξ). Since u intersects ext(γ) and ξ ⊆ int(γ), at least one of the 1-curves L(u), R(u), call it s, intersects ξ and then γ as going along s from S its basepoint. In particular, s ∈ Sk for some k ∈ {L, R}. Since ξ ⊆ H, the initial part s0 of s with respect to the skeleton (γ, Sk ) intersects a curve from H, which is therefore supported by (γ, Sk ). This contradiction the definition of H. It follows that every support of a curve from H(ξ) has both basepoints in ext(ξ).
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By Lemma 4.3, there are two consecutive intersection points a and b of ξ with the baseline such that (a, b) ⊆ int(ξ) and χ(H(ξ)(a, b)) > χ(H(ξ))/5 − 1 > 4β. The 1-curves of the curves from U that support H(ξ)(a, b) split into four families XLL , XLR ⊆ L(U) and XRL , XRR ⊆ R(U) so that XLL ∪XRL ≺ H(ξ)(a, b) ≺ XLR ∪XRR . Accordingly, H(ξ)(a, b) = HLL ∪HLR ∪HRL ∪HRR , where Hk are the curves from H(ξ)(a, b) intersecting a 1-curve from Xk for k ∈ {LL, LR, RL, RR}. Since we have χ(Hk ) > χ(H(ξ)(a, b))/4 > β for some k ∈ {LL, LR, RL, RR}, we conclude that (2) holds for Hk and Xk . When x ∈ L(F) or x ∈ R(F), we let xF denote the curve in F such that x = L(xF ) or x = R(xF ). For X ⊆ L(F) or X ⊆ R(F), we let X F = {xF : x ∈ X }. For a function h : R → R, we let h(m) denote the m-fold composition of h. (3)
Lemma 4.5. Let f (α) = g5α (4), where g is the function from Lemma 4.4. If χ(F) > f (α), then there is a subfamily G ⊆ F with χ(G) > 5α supported by a skeleton (γ, S) such that γ is contained in the union of at most 5 curves from F and S ⊆ L(F) or S ⊆ R(F). Proof. Suppose the contrary. Let F0 = F. For 1 6 i 6 3, let Fi and Si be the families obtained (3−i) by applying the conclusion (2) of Lemma 4.4 to Fi−1 . That is, Fi ⊆ Fi−1 , χ(Fi ) > g5α (4), Si ⊆ L(Fi−1 ) or Si ⊆ R(Fi−1 ), each c ∈ Fi intersects some s ∈ Si , and Fi ≺ Si or Si ≺ Fi . Two of S1 , S2 , S3 , say Si1 and Si2 where 1 6 i1 < i2 6 3, consist of 1-curves of the same type. Suppose Si2 ≺ Fi2 (the opposite case is symmetric). Let S = Si2 and X be the family of 1-curves (3−i ) of the curves from Fi2 with type opposite to Si2 . Since χ(X F ) > g5α 2 (4) > 4, we can partition X as X = Y ∪ Z so that Y ≺ Z, χ(Y F ) > 2, and χ(Z F ) > 2. For z ∈ Z, let A(z) be the region enclosed by going along z from b(z) up to the first intersection point with a 1-curve s ∈ S (to be denoted later by s(z)), then following along s to b(s), and then coming back to b(z) along the baseline. Let zˆ be the 1-curve in Z with leftmost basepoint. For every z ∈ Z, if the boundaries of A(z) and A(ˆ z ) intersect anywhere outside the baseline, then the first such intersection point as going counterclockwise from b(z) must belong to s(ˆ z ), and thus s(z) = s(ˆ z ). This shows that A(ˆ z ) ⊆ A(z) for every z ∈ Z. Since F is triangle-free, the 1-curves from Y intersecting s(ˆ z ) are pairwise disjoint, and hence removing them from Y gives a family Yˆ of 1-curves lying entirely inside A(ˆ z ) such that F F ˆ ˆ χ(Y ) > χ(Y ) − 1 > 1. Let y ∈ Y, and choose any si1 ∈ Si1 intersecting y. It follows that si1 intersects every z ∈ Z, as y ⊆ A(ˆ z ) ⊆ A(z), b(si1 ) ∈ / A(z), and si1 and s(z) cannot intersect F having the same type. This yields χ(Z ) 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 construct a triangle in the intersection graph of F under the assumption that χ(F) is large enough. If γ is a closed curve crossing the baseline, then an upper section of γ is an arc-connected component of H + r γ. Lemma 4.6. We have χ(F) < f (11) (511 K(max{20K, B} + 11)), where f is the function from Lemma 4.5 and K, 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 4.5 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 (γi , S˜i ), where γi is a closed curve contained in the union of at most 5 curves from Fi−1 . Let p1 < · · · < p2n be the intersection points of γi with the baseline. For 1 6 j 6 n, let Sij = S˜i (p2j−1 , p2j ) and Fij be the curves from F˜i
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that are supported by the skeleton (γi , Sij ). Thus S˜i = Si1 ∪· · ·∪Sin and F˜i = Fi1 ∪· · ·∪Fin . 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 = Fij and Si = Sij . Apply Lemma 4.2 to F11 to get a closed curve γ contained in the union of r 6 5 curves c1 , . . . , cr ∈ F11 such that χ(F11 (γ)) > χ(F11 )/K > 511 (max{20K, B} + 14). Let hj = L(cj ) and hr+j = R(cj ) for 1 6 j 6 r, so that the part of γ above the baseline is a subset of h1 ∪ · · · ∪ h2r . Let G0 = F11 (γ). 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 s0 of s with respect to the skeleton (γi , Si ) intersects hj , and let Gij be the family of curves from Gi−1 that have a support in Uij . Thus Si = Ui1 ∪ · · · ∪ Ui2r and Gi−1 = Gi1 ∪ · · · ∪ Gi2r . Since r 6 5 and the supports in Si can only intersect 1-curves from 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 1-curves in Ui1 . Let p1 < · · · < p2n be the intersection points of γi1 with the baseline. Recall that there is j ∈ {1, . . . , n} such that b(s1 ), . . . , b(sm ) ∈ (p2j−1 , p2j ). Let U be the upper section of int(γi1 ) containing (p2j−1 , p2j ). The initial parts s01 , . . . , s0m of s1 , . . . , sm with respect to the skeleton (γi1 , Ui1 ) are contained in U (except for the endpoints of s01 , . . . , s0m other than their basepoints). Since s01 , . . . , s0m are pairwise disjoint, the orders of the endpoints of s01 , . . . , s0m along the two parts of the boundary of U between p2j−1 and p2j are the same. Therefore, the set U r (s01 ∪ · · · ∪ s0m ) splits into m + 1 arc-connected components U0 , . . . , Um such that • the boundary of U0 is formed by s01 , [p2j−1 , b(s1 )], and the clockwise part of the boundary of U from p2j−1 to the other endpoint of s01 ; • for 1 6 k 6 m−1, the boundary of Uk is formed by s0k , s0k+1 , [b(sk ), b(sk+1 )], and the clockwise part of the boundary of U from the other endpoint of s0k to the other endpoint of s0k+1 ; • the boundary of Um is formed by s0m , [b(sm ), p2j ], and the clockwise part of the boundary of U from the other endpoint of s0m to p2j . Let X be the family of 1-curves of the type opposite to s1 , . . . , sm arising from the curves in G11 . Thus every 1-curve in X intersects s0k for some k ∈ {1, . . . , m}. The above implies that • (p2j−1 , b(s1 )) ⊆ U0 , so every 1-curve in X (p2j−1 , b(s1 )) intersects s01 ; • 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 1-curve in X (p2i−1 , p2i ) intersects s0ki or s0ki +1 ; • (b(sm ), p2j ) ⊆ Um , so every 1-curve in X (b(sm ), p2j ) intersects s0m . ˜ be the curves from G11 that do not intersect any of s0 , s0 , and s0 , s0 Let H m 1 ki ki +1 for i ∈ ˜ {1, . . . , n} r {j}. Since n 6 5, it follows that χ(H) > χ(G11 ) − 10 > max{20K, B} + 1. Now, let ˜ = s0 ∪ U1 ∪ s0 ∪ · · · ∪ Um−1 ∪ s0 , so that the boundary of U ˜ is formed by s0 , s0 , [b(s1 ), b(sm )], U m m 1 2 1 and the clockwise part of the boundary of U from the other endpoint of s01 to the other endpoint ˜ are contained of s0m . The 1-curves of the type opposite to s1 , . . . , sm arising from the curves in H ˜ . The other 1-curves of the curves in H ˜ are either contained in U ˜ or disjoint from U ˜ . Those in U ˜ ˜ ˜ curves from H whose other 1-curves are contained in U form the family H = H(b(s1 ), b(sm )), ˜ have chromatic number at most 1. Hence while those whose other 1-curves are disjoint from U ˜ χ(H) > χ(H) − 1 > max{20K, B}.
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Apply Lemma 4.2 to H to get a closed curve ζ contained in the union of at most 5 curves from H such that χ(H(ζ)) > χ(H)/K > 20. Let q1 < · · · < q2n be the intersection points of ζ with the baseline. By Lemma 4.3, there is j ∈ {1, . . . , n} such that χ(H(ζ)(q2j−1 , q2j )) > χ(H(ζ))/5 − 1 > 3. An argument analogous to the one above shows that the curves from H(ζ)(q2j−1 , q2j ) supported by the skeleton (γi2 , Ui2 r Ui2 (q2j−1 , q2j )) intersect two specific supports from Ui2 r Ui2 (q2j−1 , q2j ) and therefore have chromatic number at most 2. Since χ(H(ζ)(q2j−1 , q2j )) > 3, at least one curve from H(ζ)(q2j−1 , q2j ) must be supported by the skeleton (γi2 , Ui2 (q2j−1 , q2j )). This shows that Ui2 (q2j−1 , q2j ) 6= ∅. Let x ∈ Ui2 (q2j−1 , q2j ) and x ˜ be the 1-curve of xF opposite to x. Since b(x) ∈ (q2j−1 , q2j ) ⊆ int(ζ) and xF cannot intersect ζ below the baseline, we have b(˜ x) ∈ int(ζ). This yields b(˜ x) ∈ (b(s1 ), b(sm )), as all intersection points of ζ with the baseline lie in (b(s1 ), b(sm )). Since xF ∈ Fi1 , there is s ∈ Si1 that supports xF in the skeleton (γi1 , Si1 ). Since Si1 and x have the same type, s intersects x ˜. Recall that each 1-curve in Ui1 or Ui2 intersects h. In particular, so do s1 , sm and x. Let ξ be the closed curve formed by the interval [b(s1 ), b(sm )], 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 int(ξ) ⊆ int(γ) ⊆ int(γi1 ). Since h and x ˜ have the same type, at least one of the following holds: (1) x ˜ intersects s1 , (2) x ˜ intersects sm , or (3) x ˜ intersects s at a point in int(ξ). 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(ξ) in order to intersect γi1 , which is only possible by crossing h, because s, s1 and sm have the same type. In each of the three cases, hF , sF and xF form a triangle in the intersection graph, which is a contradiction. Proof of Theorem 1.5. Lemma 4.6 shows that triangle-free simple restricted family of curves each intersecting the baseline exactly twice have bounded chromatic number. Therefore, the assumption of Lemma 3.6 is satisfied, and Theorem 1.5 directly follows. 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.
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[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. n, Piotr Micek, Arkadiusz Pawlik, and Bartosz Walczak. Coloring intersection graphs of arc[14] Michal Laso´ connected 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:1512.06112v2 [math.CO] 9 Mar 2016
(1) each member of F crosses a fixed straight line L at least once and at most t times, (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 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) 6 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 is not χ-bounded. 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, proposed by McGuinness [16, 17], is that each curve must cross a fixed line in exactly one point. Theorem 1.2 (McGuinness [17]). Triangle-free simple families of curves each crossing 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 crossing a fixed line in exactly one point is χ-bounded. Alexandre Rok was partially supported by the Israel Science Foundation under grant 1136/12. Bartosz Walczak was partially supported by the National Science Center of Poland under grant 2015/17/D/ST1/00585. 1
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Do the statements from Theorems 1.2 and 1.3 remain valid if we allow the curves to cross the fixed line in more than one point? In view of Theorem 1.1, we cannot hope for any similar result for families of curves crossing 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, the class of intersection graphs of families of curves crossing a fixed line in at least one and at most t points is χ-bounded. Results. We present two results which offer the first progress towards resolving Conjecture 1.4. Theorem 1.5. For every t, triangle-free simple families of curves crossing 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 crossing 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 crossing 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. 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 axis-parallel 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 circle graphs—intersection graphs of families of chords of a fixed circle. Gy´arf´as [6, 7] proved that circle 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 known construction of circle graphs with large chromatic number forces Ω(ω log ω) colors [9]. McGuinness [16, 17] proposed to investigate the problem allowing much more general geometric shapes but restricting the way how they can be positioned in the plane. First, he proved that the class of intersection graphs of L-shapes crossing a fixed horizontal line is χ-bounded [16]. Then, he also showed that triangle-free simple families of curves crossing 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 crossing a fixed vertical line give rise to a χ-bounded class of graphs, and by Laso´ n et al. [14], who showed
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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 crossing 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 intersection graphs of segments or, more generally, x-monotone curves, this can be improved to Oω (log n) using the above-mentioned result of Suk [22] or Rok and Walczak [21] and 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 the construction 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 admit a curvilinear plane drawing with no k pairwise crossing edges. A well-known conjecture asserts that n-vertex k-quasi-planar graphs 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 one point and c 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 as the baseline. It is the line that curves from the families that we consider are required to cross. For any considered family of curves F, we assume that every crossing between a curve c ∈ F and 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 cross the baseline at a common point. The closed halfplanes above and below the baseline are denoted by H + and H − , respectively. The left-to-right order of the points on the baseline is denoted by 2. Suppose there is a nondecreasing function ft−1 : N → N such that every family G of curves crossing the baseline at least once and at most t − 1 times satisfies χ(G) 6 ft−1 (ω(G)). Let F be a family of curves crossing the baseline at least once and at 2 (ω(F)) such that any most t times. 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 1-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. Construct two families F1 and F2 of curves crossing the baseline at least once and at most t − 1 times, as follows: (a) for every curve c ∈ F crossing the baseline exactly t times, put c L(c) to F1 and c R(c) to F2 , (b) put every curve from F crossing the baseline less than t times to both F1 and F2 . This yields χ(Fk ) 6 ft−1 (ω(Fk )) 6 ft−1 (ω(F)) for k ∈ {1, 2}. For c ∈ F and k ∈ {1, 2}, let φk (c) be the color of the curve obtained from c in a 2 (ω(F))-coloring proper ft−1 (ω(F))-coloring of Fk . This yields a (not necessarily proper) ft−1 2 (ω(F)) and the restriction of (φ , φ ) (φ1 , φ2 ) of F. There is G ⊆ F such that χ(G) > χ(F)/ft−1 1 2 to G is constant. Since any two curves c1 , c2 ∈ F with (φ1 (c1 ), φ2 (c1 )) = (φ1 (c2 ), φ2 (c2 )) can intersect only when their opposite 1-curves intersect, G is as required in the lemma. A restricted curve is a curve c crossing the baseline such that L(c), R(c) ⊆ H + . It follows that every restricted curve crosses the baseline an even number of times. 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 ). A cap of a restricted curve c is a part of c between two consecutive basepoints that lies in H + . Every curve in a restricted family F is disjoint from all caps of the other curves in F. Two 1-curves x, y ∈ H + are separated in a restricted family F if they lie entirely in two different arc-connected components of H + r C, where C is the union of all caps of curves in 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, they lie in different arc-connected components of F r C, where C is the union of all caps of curves in F. Therefore, V and C intersect, which implies that some 1-curve of a curve from F forming the vertex V intersects some cap of a curve from F. This contradicts the assumption that F is restricted. Therefore, G has no loops. Each curve c ∈ G such that L(c) ⊆ U and R(c) ⊆ V , witnessing an edge U V of G, connects U and V in such a way that there are no intersections with other curves in between. This is
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because any such intersection would contradict the assumption that F is restricted. Therefore, G is planar, and thus χ(G) 6 4. Let φ be a proper 4-coloring of G. Color the curves in G so that the color φ(V ) of every vertex V of G is applied to every curve c ∈ G such that L(c) ⊆ V . For any c1 , c2 ∈ G, if R(c1 ) and L(c2 ) intersect, then they are part of the same vertex of G whose color is different from the color of the vertex of G containing L(c1 ). Therefore, the 4-coloring of G is proper. 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 [l(c1 ), r(c1 )] and [l(c2 ), r(c2 )] are neither nested nor disjoint, that is, l(c1 ) < l(c2 ) < r(c1 ) < r(c2 ) or l(c2 ) < l(c1 ) < r(c2 ) < r(c1 ). For k ∈ {1, 2}, since L(ck ) and R(ck ) are not separated in F, the points l(ck ) and r(ck ) can be connected by a curve cˆk ⊆ H + that avoids all caps of curves from F. We can assume without loss of generality that cˆ1 and cˆ2 cross in exactly one point p. For k ∈ {1, 2}, let γk be the closed curve obtained as the union of cˆk and the part of ck between l(ck ) and r(ck ). It follows that p is the unique crossing point of γ1 and γ2 , which contradicts the Jordan curve theorem. Lemma 3.4. If every restricted family F˜ of curves crossing the baseline exactly twice satisfies ˜ 6 f (ω(F)), ˜ χ(F) where f : N → N is a nondecreasing function, 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 − , thus obtaining a restricted family H ˜ crosses the baseline exactly twice. Therefore, intersection graph as H. Every curve in H ˜ 6 f (ω(H)) ˜ = f (ω(H)). We conclude that by the assumption of the lemma, χ(H) = χ(H) χ(F) 6 χ(G) + χ(H) 6 f (ω(H)) + 4 6 f (ω(F)) + 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 crossing the baseline exactly twice satisfies χ(F) Let f1 : N → N be the function claimed by Theorem 1.3, that is, such that every family H of curves crossing 2 (x). the baseline exactly once satisfies χ(H) 6 f1 (ω(H)). For t > 2, let ft (x) = (f (x) + 4)ft−1 Then, for t > 1, every family F of curves crossing the baseline at least once and at most t times satisfies χ(F) 6 ft (ω(F)). Proof. The proof goes by induction on t. There is nothing to prove for t = 1, so suppose 2 (ω(F)) such that any intersection t > 2. By Lemma 3.1, there is G ⊆ F with χ(G) > χ(F)/ft−1 between two curves c1 , c2 ∈ G is either between R(c1 ) and L(c2 ) or between L(c1 ) and R(c2 ). If t is odd, then no such intersection can occur, and thus χ(G) 6 1. Now, suppose t is even. Let G + = {c ∈ G : L(c), R(c) ⊆ H + } and G − = {c ∈ G : L(c), R(c) ⊆ H − }. Then G = G + ∪ G − and χ(G) = max{χ(G + ), χ(G − )}. Since G + is a restricted family, Lemma 3.4 yields χ(G + ) 6 f (ω(G + )) + 4 6 f (ω(F)) + 4. By symmetry, we also have χ(G − ) 6 f (ω(F)) + 4. We 2 (ω(F)) 6 (f (ω(F)) + 4)f 2 (ω(F)). conclude 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 crossing a fixed line at least once and at most t times gives the following reduction.
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ALEXANDRE ROK AND BARTOSZ WALCZAK
s2 s1 p1
p2 p3
p4
Figure 1. A family {p1 , p2 , p3 , p4 } externally supported by s1 and s2 Lemma 3.6. Suppose there is an integer constant α such that every triangle-free simple restricted ˜ 6 α. Let β1 be the constant family F˜ of curves crossing the baseline exactly twice satisfies χ(F) claimed by Theorem 1.2, that is, such that every triangle-free simple family H of curves crossing 2 . Then, for t > 1, the baseline exactly once satisfies χ(H) 6 β1 . For t > 2, let βt = (α + 4)βt−1 every triangle-free simple restricted family F of curves crossing the baseline at least once and at most t times satisfies χ(F) 6 βt . 4. Curves crossing the baseline 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 c ∈ G there is a curve s ∈ F that intersects c and ext(G). See Figure 1. Such a curve s is called the support of c 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 .
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A closed curve is a homeomorphic image of a circle. By the Jordan curve theorem, for every closed curve γ, the set R2 r γ splits into two arc-connected components, one bounded, called the interior of γ and denoted by int(γ), and one unbounded, called the exterior of γ and denoted by ext(γ). For a family of curves F and a closed curve γ, we let F(γ) denote the family of curves from F that lie entirely in int(γ). The following is a special case of the main result proved in [18]. Lemma 4.2 (McGuinness [18]). There are integer 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, the intersection graph of F is triangle-free, and χ(F) > B. Then there is a closed curve γ contained in the union of at most 5 curves from F such that χ(F(γ)) > 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. This is the common setting for the rest of this section—in particular, for Lemmas 4.3–4.6. Namely, we let F be a family of curves with the following properties: (1) Each curve in F crosses the baseline in exactly two points, both of which are proper crossings. (2) Both endpoints of each curve in F lie above the baseline. (3) Any intersection between two curves in F is between their 1-curves of opposite types, that is, between a right 1-curve and a left 1-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 condition (4) is that it is required by Lemma 4.2. Proving an analogue of Lemma 4.2 for more general classes of curves will automatically lead to a generalization of our Theorem 1.5. 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 that used in Section 3. Lemma 4.3. Let γ be a closed curve contained in the union of at most 5 curves from F. If G ⊆ F(γ), then there are two consecutive intersection points a, b of γ with the baseline such that χ(G(a, b)) > χ(G)/5 − 1. Proof. Let p1 < · · · < p2n be the intersection points of γ with the baseline. For 1 6 k 6 n, let Gk = {c ∈ G : l(c) ∈ (p2k−1 , p2k )}. Since γ is contained in the union of at most 5 curves each crossing the baseline twice, we have n 6 5. Therefore, for some k ∈ {1, . . . , n}, we have χ(Gk ) > χ(G)/5. It is enough to show that χ(Gk r G(p2k−1 , p2k )) 6 1. To this end, suppose c1 , c2 ∈ Gk r G(p2k−1 , p2k ) and L(c1 ) ∩ R(c2 ) = {p}. Then k < n and r(c2 ) ∈ (p2k+1 , p2n ). The subcurve of L(c1 ) from l(c1 ) to p, the subcurve of R(c2 ) from p to l(c2 ), and the interval between l(c1 ) and l(c2 ) form a closed curve contained in int(γ) whose interior contains the interval (p2k , p2k+1 ). This is a contradiction. 4.4. Rising of skeletons. A skeleton is a pair (γ, S), where γ is a closed curve, S ⊆ L(F) or S ⊆ R(F), every 1-curve in S intersects γ, and the basepoint of every 1-curve in S lies in int(γ). Since all members of S in a skeleton (γ, S) are of the same type, they are pairwise disjoint. The initial part of a 1-curve s ∈ S with respect to a skeleton (γ, S) is the part of s between b(s) and the first intersection point of s with γ. The initial part of s is denoted by s0 when the skeleton is clear from the context. A family G ⊆ F(γ) is supported by the skeleton (γ, S) if every curve c ∈ G intersects s0 for some 1-curve s ∈ S, which is called the support of c in (γ, S). See Figure 2.
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ALEXANDRE ROK AND BARTOSZ WALCZAK
p1
p2
p3
p4
Figure 2. A skeleton supporting a family {p1 , p2 , p3 } (p4 is not supported) Lemma 4.4. Let gα (β) = 2K(max{(20β + 5)K, B} + 2α), where K and B are the constants from Lemma 4.2. If χ(F) > gα (β), then one of the following holds: (1) There is a family G ⊆ F with χ(G) > α supported by a skeleton (γ, S) such that γ is contained in the union of at most 5 curves from F and S ⊆ L(F) or S ⊆ R(F). (2) There is a family G ⊆ F with χ(G) > β and a family S ⊆ L(F) or S ⊆ R(F) such that each c ∈ G intersects some s ∈ S and G ≺ S or S ≺ G. Proof. Suppose (1) does not hold. Apply Lemma 4.1 to get an externally supported subfamily G ⊆ F with χ(G) > χ(F)/2 > K(max{(20β + 5)K, B} + 2α). Next, apply Lemma 4.2 to G to get a closed curve γ contained in the union of at most 5 curves from G such that χ(G(γ)) > χ(G)/K > max{(20β + 5)K, B} + 2α. Let U denote the set of supports of G, that is, the set of curves from F intersecting ext(G) and some curve from G. Let SL = {s ∈ L(U) : b(s) ∈ int(γ)} and SR = {s ∈ R(U) : b(s) ∈ int(γ)}. Every 1-curve in SL and SR S intersects ext(γ), because it intersects ext(G) and γ ⊆ G. Removing all curves supported by at least one of the skeletons (γ, SL ), (γ, SR ) from G(γ) yields a family H ⊆ G(γ) such that χ(H) > χ(G(γ)) − 2α > max{(20β + 5)K, B}. Apply Lemma 4.2 again to H to get a closed S curve ξ ⊆ H such that χ(H(ξ)) > χ(H)/K > 20β + 5. Consider a support u ∈ U of a curve from H(ξ). Since u cannot intersect ξ below the baseline, the basepoints of u lie both in int(ξ) or both in ext(ξ). Suppose l(u), r(u) ∈ int(ξ). Then the whole subcurve of u between l(u) and r(u) lies in int(ξ). Since u intersects ext(γ) and ξ ⊆ int(γ), at least one of the 1-curves L(u), R(u), call it s, intersects ξ and then γ as going along s from S its basepoint. In particular, s ∈ Sk for some k ∈ {L, R}. Since ξ ⊆ H, the initial part s0 of s with respect to the skeleton (γ, Sk ) intersects a curve from H, which is therefore supported by (γ, Sk ). This contradiction the definition of H. It follows that every support of a curve from H(ξ) has both basepoints in ext(ξ).
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By Lemma 4.3, there are two consecutive intersection points a and b of ξ with the baseline such that (a, b) ⊆ int(ξ) and χ(H(ξ)(a, b)) > χ(H(ξ))/5 − 1 > 4β. The 1-curves of the curves from U that support H(ξ)(a, b) split into four families XLL , XLR ⊆ L(U) and XRL , XRR ⊆ R(U) so that XLL ∪XRL ≺ H(ξ)(a, b) ≺ XLR ∪XRR . Accordingly, H(ξ)(a, b) = HLL ∪HLR ∪HRL ∪HRR , where Hk are the curves from H(ξ)(a, b) intersecting a 1-curve from Xk for k ∈ {LL, LR, RL, RR}. Since we have χ(Hk ) > χ(H(ξ)(a, b))/4 > β for some k ∈ {LL, LR, RL, RR}, we conclude that (2) holds for Hk and Xk . When x ∈ L(F) or x ∈ R(F), we let xF denote the curve in F such that x = L(xF ) or x = R(xF ). For X ⊆ L(F) or X ⊆ R(F), we let X F = {xF : x ∈ X }. For a function h : R → R, we let h(m) denote the m-fold composition of h. (3)
Lemma 4.5. Let f (α) = g5α (4), where g is the function from Lemma 4.4. If χ(F) > f (α), then there is a subfamily G ⊆ F with χ(G) > 5α supported by a skeleton (γ, S) such that γ is contained in the union of at most 5 curves from F and S ⊆ L(F) or S ⊆ R(F). Proof. Suppose the contrary. Let F0 = F. For 1 6 i 6 3, let Fi and Si be the families obtained (3−i) by applying the conclusion (2) of Lemma 4.4 to Fi−1 . That is, Fi ⊆ Fi−1 , χ(Fi ) > g5α (4), Si ⊆ L(Fi−1 ) or Si ⊆ R(Fi−1 ), each c ∈ Fi intersects some s ∈ Si , and Fi ≺ Si or Si ≺ Fi . Two of S1 , S2 , S3 , say Si1 and Si2 where 1 6 i1 < i2 6 3, consist of 1-curves of the same type. Suppose Si2 ≺ Fi2 (the opposite case is symmetric). Let S = Si2 and X be the family of 1-curves (3−i ) of the curves from Fi2 with type opposite to Si2 . Since χ(X F ) > g5α 2 (4) > 4, we can partition X as X = Y ∪ Z so that Y ≺ Z, χ(Y F ) > 2, and χ(Z F ) > 2. For z ∈ Z, let A(z) be the region enclosed by going along z from b(z) up to the first intersection point with a 1-curve s ∈ S (to be denoted later by s(z)), then following along s to b(s), and then coming back to b(z) along the baseline. Let zˆ be the 1-curve in Z with leftmost basepoint. For every z ∈ Z, if the boundaries of A(z) and A(ˆ z ) intersect anywhere outside the baseline, then the first such intersection point as going counterclockwise from b(z) must belong to s(ˆ z ), and thus s(z) = s(ˆ z ). This shows that A(ˆ z ) ⊆ A(z) for every z ∈ Z. Since F is triangle-free, the 1-curves from Y intersecting s(ˆ z ) are pairwise disjoint, and hence removing them from Y gives a family Yˆ of 1-curves lying entirely inside A(ˆ z ) such that F F ˆ ˆ χ(Y ) > χ(Y ) − 1 > 1. Let y ∈ Y, and choose any si1 ∈ Si1 intersecting y. It follows that si1 intersects every z ∈ Z, as y ⊆ A(ˆ z ) ⊆ A(z), b(si1 ) ∈ / A(z), and si1 and s(z) cannot intersect F having the same type. This yields χ(Z ) 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 construct a triangle in the intersection graph of F under the assumption that χ(F) is large enough. If γ is a closed curve crossing the baseline, then an upper section of γ is an arc-connected component of H + r γ. Lemma 4.6. We have χ(F) < f (11) (511 K(max{20K, B} + 11)), where f is the function from Lemma 4.5 and K, 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 4.5 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 (γi , S˜i ), where γi is a closed curve contained in the union of at most 5 curves from Fi−1 . Let p1 < · · · < p2n be the intersection points of γi with the baseline. For 1 6 j 6 n, let Sij = S˜i (p2j−1 , p2j ) and Fij be the curves from F˜i
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ALEXANDRE ROK AND BARTOSZ WALCZAK
that are supported by the skeleton (γi , Sij ). Thus S˜i = Si1 ∪· · ·∪Sin and F˜i = Fi1 ∪· · ·∪Fin . 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 = Fij and Si = Sij . Apply Lemma 4.2 to F11 to get a closed curve γ contained in the union of r 6 5 curves c1 , . . . , cr ∈ F11 such that χ(F11 (γ)) > χ(F11 )/K > 511 (max{20K, B} + 14). Let hj = L(cj ) and hr+j = R(cj ) for 1 6 j 6 r, so that the part of γ above the baseline is a subset of h1 ∪ · · · ∪ h2r . Let G0 = F11 (γ). 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 s0 of s with respect to the skeleton (γi , Si ) intersects hj , and let Gij be the family of curves from Gi−1 that have a support in Uij . Thus Si = Ui1 ∪ · · · ∪ Ui2r and Gi−1 = Gi1 ∪ · · · ∪ Gi2r . Since r 6 5 and the supports in Si can only intersect 1-curves from 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 1-curves in Ui1 . Let p1 < · · · < p2n be the intersection points of γi1 with the baseline. Recall that there is j ∈ {1, . . . , n} such that b(s1 ), . . . , b(sm ) ∈ (p2j−1 , p2j ). Let U be the upper section of int(γi1 ) containing (p2j−1 , p2j ). The initial parts s01 , . . . , s0m of s1 , . . . , sm with respect to the skeleton (γi1 , Ui1 ) are contained in U (except for the endpoints of s01 , . . . , s0m other than their basepoints). Since s01 , . . . , s0m are pairwise disjoint, the orders of the endpoints of s01 , . . . , s0m along the two parts of the boundary of U between p2j−1 and p2j are the same. Therefore, the set U r (s01 ∪ · · · ∪ s0m ) splits into m + 1 arc-connected components U0 , . . . , Um such that • the boundary of U0 is formed by s01 , [p2j−1 , b(s1 )], and the clockwise part of the boundary of U from p2j−1 to the other endpoint of s01 ; • for 1 6 k 6 m−1, the boundary of Uk is formed by s0k , s0k+1 , [b(sk ), b(sk+1 )], and the clockwise part of the boundary of U from the other endpoint of s0k to the other endpoint of s0k+1 ; • the boundary of Um is formed by s0m , [b(sm ), p2j ], and the clockwise part of the boundary of U from the other endpoint of s0m to p2j . Let X be the family of 1-curves of the type opposite to s1 , . . . , sm arising from the curves in G11 . Thus every 1-curve in X intersects s0k for some k ∈ {1, . . . , m}. The above implies that • (p2j−1 , b(s1 )) ⊆ U0 , so every 1-curve in X (p2j−1 , b(s1 )) intersects s01 ; • 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 1-curve in X (p2i−1 , p2i ) intersects s0ki or s0ki +1 ; • (b(sm ), p2j ) ⊆ Um , so every 1-curve in X (b(sm ), p2j ) intersects s0m . ˜ be the curves from G11 that do not intersect any of s0 , s0 , and s0 , s0 Let H m 1 ki ki +1 for i ∈ ˜ {1, . . . , n} r {j}. Since n 6 5, it follows that χ(H) > χ(G11 ) − 10 > max{20K, B} + 1. Now, let ˜ = s0 ∪ U1 ∪ s0 ∪ · · · ∪ Um−1 ∪ s0 , so that the boundary of U ˜ is formed by s0 , s0 , [b(s1 ), b(sm )], U m m 1 2 1 and the clockwise part of the boundary of U from the other endpoint of s01 to the other endpoint ˜ are contained of s0m . The 1-curves of the type opposite to s1 , . . . , sm arising from the curves in H ˜ . The other 1-curves of the curves in H ˜ are either contained in U ˜ or disjoint from U ˜ . Those in U ˜ ˜ ˜ curves from H whose other 1-curves are contained in U form the family H = H(b(s1 ), b(sm )), ˜ have chromatic number at most 1. Hence while those whose other 1-curves are disjoint from U ˜ χ(H) > χ(H) − 1 > max{20K, B}.
COLORING CURVES THAT CROSS A FIXED LINE
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Apply Lemma 4.2 to H to get a closed curve ζ contained in the union of at most 5 curves from H such that χ(H(ζ)) > χ(H)/K > 20. Let q1 < · · · < q2n be the intersection points of ζ with the baseline. By Lemma 4.3, there is j ∈ {1, . . . , n} such that χ(H(ζ)(q2j−1 , q2j )) > χ(H(ζ))/5 − 1 > 3. An argument analogous to the one above shows that the curves from H(ζ)(q2j−1 , q2j ) supported by the skeleton (γi2 , Ui2 r Ui2 (q2j−1 , q2j )) intersect two specific supports from Ui2 r Ui2 (q2j−1 , q2j ) and therefore have chromatic number at most 2. Since χ(H(ζ)(q2j−1 , q2j )) > 3, at least one curve from H(ζ)(q2j−1 , q2j ) must be supported by the skeleton (γi2 , Ui2 (q2j−1 , q2j )). This shows that Ui2 (q2j−1 , q2j ) 6= ∅. Let x ∈ Ui2 (q2j−1 , q2j ) and x ˜ be the 1-curve of xF opposite to x. Since b(x) ∈ (q2j−1 , q2j ) ⊆ int(ζ) and xF cannot intersect ζ below the baseline, we have b(˜ x) ∈ int(ζ). This yields b(˜ x) ∈ (b(s1 ), b(sm )), as all intersection points of ζ with the baseline lie in (b(s1 ), b(sm )). Since xF ∈ Fi1 , there is s ∈ Si1 that supports xF in the skeleton (γi1 , Si1 ). Since Si1 and x have the same type, s intersects x ˜. Recall that each 1-curve in Ui1 or Ui2 intersects h. In particular, so do s1 , sm and x. Let ξ be the closed curve formed by the interval [b(s1 ), b(sm )], 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 int(ξ) ⊆ int(γ) ⊆ int(γi1 ). Since h and x ˜ have the same type, at least one of the following holds: (1) x ˜ intersects s1 , (2) x ˜ intersects sm , or (3) x ˜ intersects s at a point in int(ξ). 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(ξ) in order to intersect γi1 , which is only possible by crossing h, because s, s1 and sm have the same type. In each of the three cases, hF , sF and xF form a triangle in the intersection graph, which is a contradiction. Proof of Theorem 1.5. Lemma 4.6 shows that triangle-free simple restricted family of curves each intersecting the baseline exactly twice have bounded chromatic number. Therefore, the assumption of Lemma 3.6 is satisfied, and Theorem 1.5 directly follows. 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.
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ALEXANDRE ROK AND BARTOSZ WALCZAK
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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]
