ON THE CHOICE NUMBER OF RANDOM ... - CiteSeerX
ON THE CHOICE NUMBER OF RANDOM HYPERGRAPHS
Van H. Vu Department of Mathematics, Yale University 10 Hillhouse, New Haven, CT-06520 May 18, 1998
Abstract.
We generalize the notion of choice number from graphs to hypergraphs and estimate the sharp order of magnitude of the choice number of random hypergraphs. It turns out that the choice number and the chromatic number of a random hypergraph have the same order of magnitude, almost surely. Our result implies an earlier bound on the chromatic number of random hypergraphs, proved by Schmidt [Sch] using a different method.
§1 INTRODUCTION
¡ ¢ The k-uniform random hypergraph H(k, n, p) is obtained by choosing each of nk kelement subsets of a vertex set V (|V | = n) independently with probability p. The chosen subsets are the hyperedges of the hypergraph. Two vertices are adjacent if there is a hyperedge containing both of them. For k = 2, we have the usual definition of the random graph G(n, p) (see [Bol], for instance). In the whole paper, k is a fixed constant, and n tends to infinity. We say that H(k, n, p) has a property P almost surely, if the probability that H(k, n, p) satisfies P is 1 − o(1). Moreover, if the probability that P does not hold is of order O(exp(−ω(log n)), that is, super-polynomially small, then we say that H(k, n, p) has P with very high probability . This notion gives more accurate information about the rate of decay of the error term o(1), which could ¡n¢ be important in applications. We identify H(k, n, p) with the space S generated by k i.i.d (0, 1) random variables tA ’s with mean p, A ∈ A, where A is the family of all subsets of size k of V . The points of this space correspond to the k-uniform hypergraphs on V , and the probability that H(k, n, p) has a property P is equal to the measure of the subset (of S) consisting of those points (hypergraphs) which satisfy P . In this paper, we make use of standard asymptotic notations such as o, O, ω, Ω and Θ. Readers unfamiliar to these notations can check [Bol2] or [AS] for definitions. Given a graph G, a proper coloring of G is a coloring such that no two adjacent vertices receive the same color. The chromatic number of G, χ(G), is the smallest number of colors one needs to use to color G properly. This notion could be generalized for hypergraphs in many ways. In this paper, we will concentrate on the following definition, which generalizes the above notion of proper coloring in, perhaps, the most natural manner. Definition 1.1 A hypergraph H is properly colored (in strong sense) if no two adjacent vertices receive the same color. The strong chromatic number of H, χ(H), is the smallest number of colors one needs to use to properly color H.
2
VAN H. VU
The problem of determining the chromatic number χ(G(n, p)) of the random graph G(n, p) is very well known and was considered by several researchers. Results of Bollob´as [Bol] [Bol2], L Ã uczak [ÃLuc], Matula [Mat], Grimmett and McDiarmid [GM] show that Theorem 1.2. There are positive constants c1 and c2 such that almost surely c1 np/ log(np) < χ(G(n, p)) < c2 np/ log(np). The sharp values of the constants c1 and c2 were determined by Bollob´as [Bol2] for large p, and by L Ã uczak [ÃLuc] for small p, so the problem is completely solved. To generalize Theorem 1.2, various authors studied the chromatic number of random hypergraphs [Sch], [SSU], [Sha]. Althought the above notion of (strong) chromatic number of hypergraphs is a natural generalization of that of graphs, the problem is more complex and requires new ideas. The sharp order of magnitude of χ(H(k, n, p)) was found by Schmidt [Sch] Theorem 1.3. Suppose p = p(n) is such a function of n that d(n) = nk−1 p/(k − 2)! tends to infinity but d(n) = O(n). Then there exists positive constants c1 and c2 such that almost surely c1 d(n)/ log(d(n)) < χ(H(k, n, p)) < c2 d(n)/ log(d(n)) i.e, χ(H(k, n, p)) = Θ(d(n)/ log(d(n))). Theorem 1.3 generalizes Theorem 1.2 (if k = 2 then d(n) = np); however, in [Sch] it was not clear what the sharp values of c1 and c2 are , when k > 2. The computation in [Sch] showed that one can set c1 ∼ 1/2 and c2 ∼ 2. In [Sha], Shamir proved that if d(n) > n6/7 , then c1 ∼ c2 ∼ 1/2. This result was extended to all values of d(n) in a recent paper of Krivelevich and Sudakov [KS]. To describe our goal, we now introduce the notion of choice number of hypergraphs. The notion of choice number of graphs was introduced by Erd˝os, Rubin and Taylor in [ERT], and also by Vizing in [Viz]. Assign to each vertex v in a graph G a list Lv of t colors (different vertices may have different lists), a list coloring is a coloring in which every vertex is colored by a color from its own list. The choice number χl (G) of the graph is the least number t such that there exists a proper list coloring for every assignment of lists of size t to the vertices. It is fairly trivial that χl (G) ≥ χ(G) for all G. Recently, choice number becomes an exciting research topics, leading to many beautiful questions and theorems (see [Alo] for a survey). Following Definition 1.2, we now define the (strong) choice number of a hypergraph. Definition 1.4. Given a hypergraph H, the choice number of H, χl (H), is the smallest t such that H admits a proper coloring for every possible assignment of lists of size t. The huge number of possible assignments makes the problem of estimating choice numbers much more complex than that of chromatic numbers. It is already hard to compute the choice number in relatively simple graphs, for which the chromatic numbers are easily seen. Let us, for instance, consider the complete bipartite graph Kn,n . While its chromatic number is 2, we do not know the exact value of its choice number. Furthermore, it was proven by Erd˝os, Rubin and Taylor that χl (Kn,n ) = (1 + o(1)) log n [ERT]. This shows that the ratio χl (G)/χ(G) (which is lower bounded by 1) can be arbitrarily large. As the
ON THE CHOICE NUMBER OF RANDOM HYPERGRAPHS
3
worst case analysis is concerned, it is shown in [ERT] that computing the choice number of a given graph is strictly harder than computing the chromatic number, under a commonly believed hypothesis that coN P 6= N P . Althought in many special graphs, the choice number could be much larger than the chromatic number, it turns out that for a typical graph, they have the same order of magnitude. The present author [Vu] showed that almost surely χl (G(n, p) = Θ(c(²)np/ log(np)) for 0.9 > p > n−1 log1+² n with any positive constant ². In [AKS], Alon, Krivelevich and Sudakov showed that Θ(np/ log(np)) is the right order of magnitude of χl (G(n, p)), for all p such that 2 ≤ np ≤ n/2. Together, we have Theorem 1.5. There are positive constants c1 and c2 such that for all p, 2 ≤ np ≤ 0.9n, almost surely c1 np/ log(np) < χl (G(n, p)) < c2 np/ log(np). In this paper, we determine the sharp order of magnitude of the choice number of random hypergraphs. Our main theorem (generalizing Theorem 1.5) shows that almost surely, the choice number and the chromatic number of a random (k-uniform) hypergraph have the same order of magnitude (the order of magnitude of χ(H(k, n, p)) was shown in Theorem 1.3). Main Theorem. Assume that k > 2, there are positive constants c1 and c2 such that c1 d(n)/ log d(n) < χl (H(k, n, p)) < c2 d(n)/ log d(n) almost surely, where d(n) = nk−1 p/(k − 2)! and d(n) = O(n). Moreover, when d(n) = ω(1), the above statement is true with very high probability. If k = 2, the same result holds under an additional assumption that d(n) < 0.9n. The Main Theorem implies Schmidt’s theorem (and therefore it also implies Theorem 1.2), with somewhat more generous constants. On the other hand, it gives a stronger statement (compared to Theorems 1.3 and 1.5) in the sense that it shows if d(n) tends to infinity then the probability that the choice number has wrong order of magnitude not only tends to 0, but does it super-polynomially fast. This guarantees that if one randomly picks polynomially many hypergraphs, then with very high probability their (strong) choice numbers (or chromatic numbers) differ by only a constant factor, given d(n) → ∞. Such property might be useful for applications in computer science, especially in the area of randomized algorithms, where one often has to make (usually polynomially many) random samples from some space. In the next section, we discuss the main difficulties concerning choice number and random hypergraphs, and present the main idea of the proof and some key tools. The proof of Main Theorem appears in Section 3. In Section 4, we discuss the algorithmic aspect of Main Theorem. We show that for most values of p, there is a polynomial time (randomized) algorithm, by which one can properly color almost every hypergraph in H(k, n, p) using any assignment of lists of size c2 d(n)/ log d(n). We conclude with Section 5, which contains few remarks and open questions. §2 PREMILINARIES A set of vertices is independent if its vertices are pairwise non-adjacent. Given a hypergraph H, we denote by α(H) the cardinality of the largest independent set of H. It is clear that χ(H) ≥ n/α(H).
4
VAN H. VU
The lower bound in Theorem 1.3 (and Theorem 1.2) is proved by a simple “first moment” argument, which shows that almost surely, H(k, n, p) cannot have a “big” independent set with more than cn log(d(n))/d(n) vertices, for some constant c (see [Bol], [Sch], for instance). It is easily seen that if d(n) = ω(1), then for appropriate c, the probability of having a “big” independent set is super-polynomially small. Since the choice number is at least the chromatic number, this implies the lower bound in Main Theorem. The more involved part of the proof of Main Theorem is to show the upper bound. Let us first discuss the difficulties one may face when one needs to estimate the choice number instead of the chromatic number. The general idea in the proofs of the upper bounds for chromatic numbers in Theorems 1.3 and 1.2 is to show that almost surely there are many disjoint independent sets of proper order (Θ(n log d(n)/d(n))), whose union covers almost every vertex. The rest (and negligible) part of the vertex set can be colored greedly afterwards. Unfortunately, this technique does not always help to give an upper bound for the choice number. Even when the graph can be covered by only two independent sets (as shown in the example of Kn,n ), the choice number could be arbitrarily large. Therefore, we need to find a different approach to tackle the problem. Before showing the idea, let us point out another potential obstruction. Consider a hypergraph H on a set V of n elements. Let GH denote the graph on V obtained by joining all pairs (x, y), where x and y are adjacent in H. Following the definition of choice number of hypergraphs, it is easy to see that χl (H) = χl (GH ). Readers may think that by this simple trick, the problem on hypergraphs is reduced to graphs, and one could apply Theorem 1.5 to prove Main Theorem. However, the situation is not so simple; while H(k, n, p) is a random hypergraph, it is not true that GH(k,n,p) is a random graph. The fact that GH(k,n,p) is not random turns out to be a serious obstruction in the investigation of many properties and makes the study of random hypergraphs considerably more involved than that of random graphs (see [KL] for a collection of important, unsolved problems concerning random hypergraphs). Let us, for instance, consider the problem of computing α(H(k, n, p)) = α(GH(k,n,p) ). For k = 2 (GH(k,n,p) = G(n, p)), it is quite easy to compute α(G(n, p)) up to a 1 + o(1) multiplicative factor [Bol] [AS], while for k > 2 such result was obtained just very recently [KS]. The reason is that when k = 2, the property that a subset V 0 of GH(k,n,p) is independent is determined by the edges contained in V 0 . So if V 0 and V ” are disjoint, then the events that V 0 and V ” are independent sets are probabilistically independent, which makes the computation easy. On the other hand, if k > 2, the property that V 0 is independent is determined by the hyperedges intersecting V 0 in at least 2 vertices, and the above mentioned feature is lost. The key idea, which helps us to handle these obstructions is the following observation, pointed out by the present author in [Vu]: The desired upper bound on the choice number of a graph can be derived from some local properties. Let us describe what we mean by this observation. Consider two vertices x and y, the degree of x is the number of vertices adjacent to x, while the codegree of x and y is the number of vertices adjacent to both x and y. In a random graph G(n, p), the expected value of a degree is roughly np, and that of a codegree is roughly np2 . It is shown that if in a deterministic graph G, the degrees and codegrees are “not too far” from these values (np and np2 , resp), then the choice number of G is O(np/ log(np)), which is the bound we desire. This observation, at one strike, helps us to solve both of the above mentioned difficulties.
ON THE CHOICE NUMBER OF RANDOM HYPERGRAPHS
5
First, it gives a new approach to prove the desired bound on the choice number. Moreover, since the condition involves only degrees and codegrees, it is relatively easy to check, even in GH(k,n,p) , which is not a random graph in the traditional sense. The proof of Main Theorem consists essentially of the verification of the degree-codegree condition, which varies depending on the order of magnitude of p. Let us now present few theorems to support the above observation. In Section 4, we will sketch the proofs. For complete proofs, we refer to [Vu]. Theorem 2.1. Let γ be a positive constant less than 1/2. Assume that p, ² and ²0 satisfy 2γ−1
0.9 > p = ω(n 3−2γ ) (np)γ p−1 (² + ²0 ) = o(1). Consider a graph G on n vertices. Assume that every degree in G is between np(1 + ²) and np(1 − ²), and every codegree is between np2 (1 + ²0 ) and np2 (1 − ²0 ), respectively. Then χl (G) ≤ c
np γ log(np)
for some absolute constant c. Theorem 2.2. Assume that n−1 < p < n−² for some positive constant ². If G is a graph on n vertices with every degree at most anp and every codegree at most bnp2 , for some constants a and b, then there is a constant c such that χl (G) ≤ cnp/ log(np). Theorem 2.2 could be strengthened in the following way: first one can relax the condition on codegrees, allowing a larger multiplicative error term. Secondly, instead of requiring the codegree condition for every pair x and y, it is enough to keep this condition on those x and y where x and y are adjacent. To make it easier to apply, in the following Theorem we replace np by ∆. Theorem 2.3. Let α be a positive constant less than 1. Assume G is a graph on n vertices with maximal degree at most ∆, and every edge of G is contained in at most ∆1−α triangles. Then χl (G) ≤ c
∆ log(∆)
for some constant c. Remark. In [AKS], a theorem similar to Theorem 2.1 is proven (Theorem 1.2 in [AKS]). This theorem gives the same bound for the choice number under weaker conditions (it needs only one-side bounds on the degrees and codegrees), but for a smaller range of p (it is valid 1 for p > n−1/12 , while Theorem 2.1 is valid for p > n− 3 +² for any ² > 0). We will also need the following technical Lemmas. The first Lemma is a well known Theorem due to Azuma, which is very useful in proving strong concentration of functions satisfying the Lipchitz condition.
6
VAN H. VU
Lemma 2.4. (Azuma theorem) Let 0 = X1 , . . . , Xm be a martingale with |Xi+1 − Xi | ≤ 1 for all 0 ≤ i < m. Then for any λ > 0, 2
P r(Xm > λm1/2 ) < e−λ
/2
.
The martingale we consider in this paper is the so-called edge exposure martingale (for the general set up we refer to [AS]). Lemma 2.4 implies (with room to spare) that if F is a (hyper)-graphtheoretic function, and changing one (hyper)-edge changes the value of F by at most a constant, then with probability 1 − exp(−Ω(log2 m)), F has value between E(F ) ± E(F )1/2 log m. In the proof, we will also need to show strong concentration of certain multi-variable polynomials, which do not satisfy the Lipchitz condition (changing one variable may change the value of the function significantly). The key tool to handle these cases is Lemma 2.5, which is a special case of a more general Lemma in [KV]. To state Lemma 2.5, we need some preparation. Let K be a r-uniform hypergraph with V (K) = {1, 2, ..., n}. Suppose ti , i = 1, , 2..., n are i.i.d. (0, 1) random variables and E(ti ) = p. Consider the following polynomial X Y
YK =
ts
e∈E(K) s∈e
where E(K) is the edge set of K. For each subset A of V (K), |A| ≤ r, define a hypergraph KA as follows • V (KA ) = V (K)\A, • E(KA ) = {B ⊂ V (KA ), B ∪ A ∈ E(K)}. Now let Ei (K) = maxA⊂V (K),|A|=i E(YKA ). It is clear that E0 (K) = E(YK ). Furthermore, let E = maxi≥0 Ei and E 0 = maxi≥1 Ei . Lemma 2.5. (Hypergraph Lemma) There exist positive numbers cr , dr depending only on r so that for any λ > 0 P r(|YK − E0 (K)| > cr (EE 0 )1/2 λr ) < dr exp(−λ + (r − 1) log n). In particular, when r is small (r ≤ 5, say) P r(|YK − E0 (K)| > (EE 0 )1/2 logr+1 n) = O(exp(−ω(log n))). Lemma 2.5 and its more general version in [KV] have many interesting applications, especially in situations where Azuma theorem cannot be applied, because of the failure of the Lipchitz condition. These applications will be discussed in another paper. The next two Lemmas reveal some useful information about GH(k,n,p) . In all proofs in the next Section and the remaining of this Section, we assume that d(n) = pnk−1 /(k − 2)! is sufficiently large, whenever needed. Lemma 2.6. Almost surely (1) For every X ⊂ V , |X| ≤ 2n/log 2 d(n) the induced subgraph of GH(k,n,p) spanned by X has less than (d(n)/ log d(n))|X| edges.
ON THE CHOICE NUMBER OF RANDOM HYPERGRAPHS
7
(2) All but at most n/ log2 d(n) vertices of G have degree less than 10kd(n). Moreover, if d(n) = ω(1), both statements hold with very high probability. Proof. This Lemma was proved in [AKS] for the special case k = 2, i.e., GH(k,n,p) is the random graph G(n, p). Although for k > 2, GH(k,n,p) is not random, somewhat similar computation could still be carried out. Consider a set X, denote by F (X) the number of hyperedges of H(k, n, p) which intersects X in at least two points. It is clear that k 2 F (X) is larger than the number of edges of the subgraph spanned by X (in GH(k,n,p) ). Similarly, let G(X) denote the number of hyperedges which intersects X in exactly one points. Set r = d(n)/k 2 log(d(n)), s = 2n/ log2 d(n), and assume |X| = i ≤ s. Without loss of generality we may assume that r and s are positive integers. We have µ¡ i ¢¡ n ¢¶ P r(F (X) > ri) < 2 k−2 pri ri ≤(
ei2 nk−2 ri ri eid(n) ri eik 2 log d(n) ri ) p (10ks0 d(n) − s0 d(n))/k > 9s0 d(n) = t. Therefore, X is a special set. Since the probability that a special set exists is o(1), we complete the proof of part (2) of the Lemma. To prove the “with very high probability” part of (1), it suffices to notice that if d(n) = ω(1), the last sum in (2.1) is super-polynomially small in n. Indeed, it is easy to show that the terms of this sum are decreasing, and the first term is super-polynomially small. Since there are less than n terms, we are done. The “with very high probability” part of (2) follows from the fact that (1) holds with very high probability (given d(n) = ω(1)) and the rightmost formula in (2.2) is super-polynomially small. ¤
8
VAN H. VU
Lemma 2.7. Assume that d(n) < log4 n, then with very high probability, there is a set X of o(log17 n) points such that in the subgraph of GH(k,n,p) spanned by V \X, each edge is contained in at most k triangles. Proof. Let A, B, C be three edges of the random hypergraph. We say that the triple {A, B, C} is bad if there are three different points x, y, z such that x ∈ A ∩ B, y ∈ A ∩ C and z ∈ B ∩ C. We write {x, y, z} ◦ {A, B, C} if x, y, z and A, B, C satisfy the above conditions. Furthermore, we say that the pair {A, B} is bad if |A ∩ B| ≥ 2. First we show that with very high probability the numbers of bad pairs and bad triples are small. It is easily checked that the number of bad pairs (as a random variable) is at most X
X
tA tB = Y.
x,y∈V {x,y}∈A∩B x6=y A6=B
Let us first estimate the expected value of Y µ ¶ µ ¶ n n E(Y ) ∼ (p )2 < d(n)2 ≤ log8 n. 2 k−2 It follows immediately from Markov inequality that P r(Y > log11 n) = O(1/ log3 n) = o(1), that is, almost surely there are at most log11 n bad pairs. However, we want to prove a stronger statement that the same event holds with very high probability, i.e, P r(Y > log11 n) is not only o(1), but super-polynomially small. Azuma theorem cannot be applied for this purpose, since Y has very large Lipchitzian coefficient (at ¡certain ¢ points, n−2 changing the value of one variable may change the value of Y as much as k−2 ). A very convenient tool for this situation is the Hypergraph Lemma (Lemma 2.5). Notice that the polynomial Y has degree r = 2. Moreover, E0 (Y ) = E(Y ) < log8 n and E2 (Y ) = O(1) and µ ¶ µ ¶ k n E1 (Y ) ≤ p = O(d(n)/n) = o(1). 2 k−2 So Hypergraph Lemma implies that with very high probability Y ≤ log11 n,i.e, with very high probability there are less than log11 n bad pairs. Now let us consider the number of bad triples. This number is upper bounded by X
X
x,y,z∈V
A,B,C∈A {x,y,z}◦{A,B.C}
tA tB tC = Y 0 .
In the summation we assume that x, y, z ( A, B, C) are different points (subsets), resp. In order to use Hypergraph Lemma, let us notice that E3 (Y 0 ) = O(1). Moreover, µ ¶ µ ¶ n n E(Y ) = E0 (Y ) ∼ (p )3 < d(n)3 ≤ log12 n, 3 k−2 0
0
µ 0
2
E1 (Y ) = O(1)p
¶µ
n k−1
¶
n k−2
= o(1),
ON THE CHOICE NUMBER OF RANDOM HYPERGRAPHS
9
¶ µ n E2 (Y ) = O(1)p = o(1). k−2 0
Since the polynomial has degree 3, Hypergraph Lemma thus implies that with very high probability Y 0 < log16 n, that is, the number of bad triples is less than log16 n with very high probability. Let X be the set of vertices covered by a bad pair or a bad triple, then |X| = o(log17 n). Setting W = V \X, one could prove without any difficulty that the induced subgraph spanned by W satisfies the statement of the Lemma. ¤ The last Lemma is deterministic, and was used in [AKS]. A similar statement was proven earlier in [ÃLuc]. Lemma 2.8. Let G = (V, E) be a graph on n vertices such that there is a subset U0 ∈ V of size (1 + o(1))n/ log2 d(n) such that the induced subgraph spanned by V \U0 has choice number of order O(d(n)/ log d(n)). Suppose further that every q < 2n/ log2 d(n) vertices of G spanned less than qd(n)/ log d(n) edges. Then χl (G) = O(d(n)/ log d(n)). §3 THE PROOF OF MAIN THEOREM Case 1: d(n) ≥ n0.9 . This case can be solved by applying Theorem 2.1 (with suitable parameters). To do this, we first need to estimate the degrees and codegrees of GH(k,n,p) . Consider a vertex n−2 x. For any y 6= x, the probability that x and y are not adjacent is (1 − p)(k−2) = q . Let 1
p1 = 1 − q1 , the expected value of number of neighbors of x is p1 n. It is easy to check that p1 n = Θ(d(n)), so we may assume that p1 > n−0.11 . Observe that dx is a random variable depending on the atom variables tA ’s, and changing the value of one of the tA ’s will change the value of dx by at most k − 1. So, by Azuma theorem, we have with very high probability that p1 n − n0.6 < p1 n − log2 n(p1 n)1/2 < dx < p1 n + log2 n(p1 n)1/2 < p1 n + n0.6 . It follows that with very high probability, all degrees in GH(k,n,p) are between p1 n − n and p1 n + n0.6 . Now consider the codegree dx,y of x and y. First we need to estimate the expected value of dx,y . Let M be expected value of the number ¡ ¢ of paths of length 3 in GH(k,n,p) ; by symmetry the expected value of dx,y equals M/ n2 . On the other hand, with very high probability, each vertex in GH(k,n,p) has degree between d1 = p1 n − n0.6 and d2 = p1 n + n0.6 , so¡ with very ¢ ¡ ¢high probability, the number of paths of length 3 in GH(k,n,p) is between n d21 and n d22 . This implies 0.6
n((p1 n)2 /2 − 2n1.6 ) < M < n((p1 n)2 /2 + 2n1.6 ). So p21 n − 4n0.6 < E(dx,y ) < p21 n + 4n0.6 . Using Azuma theorem (Lemma 2.4), one can show that with very high probability, p21 n − 5n0.6 < dx,y < p21 n + 5n0.6 . It follows that with very high probability, all codegrees of GH(k,n,p) are between p21 n − 5n0.6 and p21 n + 5n0.6 . To apply Theorem 2.1, set ² = ²0 = n−0.15 , γ = 0.01. Notice that since p1 > n−0.11 , the degrees and codegrees of GH(k,n,p) satisfy the conditions of Theorem 2.1 (with p1 playing the role of p), with very high probability. Therefore, by Theorem 2.1, we
10
VAN H. VU
have with very high probability χl (GH(k,n,p) ) = O(np1 / log(np1 )) = O(d(n)/ log d(n)), completing the proof. Case 2: log4 n < d(n) < n0.9 . Consider the graph GH(k,n,p) . Let us first estimate the degrees in GH(k,n,p) . For a point x ∈ V , let dx denote the degree of x in GH(k,n,p) , we have
(3.1)
dx ≤ (k − 1)
X
tA = Y.
x∈A,|A|=k
¡ n ¢ The expected value of Y is (k − 1)p k−1 ∼ d(n). Since this is a sum of independent variables, it is easy to show (via Chernoff bound) that with very high probability Y ≤ 2d(n) (here we need the assumption that d(n) > log4 n; if d(n) is too small, d(n) = log log n, for instance, the statement is not true). Consider a pair of vertices (x, y). Similarly to (3.1), we have
(3.2)
dx,y ≤
X
(
X
z∈V \{x,y} x,z∈A,|A|=k
tA )(
X
tB ) = Y 0 .
y,z∈B,|B|=k
¡ n ¢2 The expected value E(Y 0 ) = E0 (Y 0 ) of Y 0 is roughly n(p k−2 ) ∼ d(n)2 /n. To use ¡ ¢ n Hypergraph Lemma, notice that E1 (Y 0 ) = O(p k−2 ) = o(1) and E2 (Y 0 ) = O(1). Hypergraph Lemma yields that with very high probability Y < c(E(Y 0 ) + 1) log3 n, for some constant c. It is easy to check that c(E(Y 0 ) + 1) log3 n = o(d(n)0.9 ) for all d(n) in the interval of interest. ¡ ¢ Since there are only n2 pairs (x, y), we conclude that with very high probability every degree in GH(k,n,p) is at most 2d(n), and every codegree in GH(k,n,p) is less than d(n)0.9 . Set ∆ = 2d(n). It follows directly from Theorem 2.3 that there is a constant c such that with very high probability χl (GH(k,n,p) ) ≤ c∆/ log ∆ = cd(n)/ log d(n), which completes the proof for this case. Case (3): d(n) ≤ log4 n. One cannot repeat the argument used in the previous case in this case, because when d(n) is too small, the strong concentration results do not hold. The trick here is to “divide and conquer”, which was applied earlier for random graphs in [AKS]. Let U be the subset of V which contains all those vertices, which either have degree at least 10kd(n) or are covered by a bad pair or bad triple. Due to Lemma 2.7 and the second part of Lemma 2.6 , |U | < n/ log2 d(n) + log17 n = (1 + o(1))n/ log2 d(n) almost surely. Let W = V \U , and G0 the subgraph spanned by W . By the definition of U , the maximum degree of G0 is at most 10kd(n) almost surely. Moreover, by Lemma 2.7, almost surely, each edge in G0 is contained in at most k triangles. Without loss of generality, one can assume that d(n) > k. Set ∆ = 10kd(n), then the condition of Theorem 2.3 is satisfied by G0 almost surely. Therefore, by Theorem 2.3, there is a constant c such that the choice number of G0 is at most c∆/ log ∆ = cd(n)/ log d(n) almost surely Applying Lemma 2.8, we can conclude that almost surely χl (GH(k,n,p ) ≤ cd(n)/ log d(n), for some constant c.
ON THE CHOICE NUMBER OF RANDOM HYPERGRAPHS
11
To show that the bound holds with very high probability (given d(n) = ω(1)), it suffices to observe that under this assumption the statements in Lemma 2.6 hold with very high probability. ¤ §4 THE ALGORITHMIC ASPECT In this Section, we study the algorithmic aspect of Main Theorem. Let us first start with the special case of random graph and chromatic number. Theorem 1.2 states that the random graph G(n, p) has chromatic number at most cnp/ log np almost surely, for some constant c. We consider the following question Is there a polynomial algorithm which could color almost every graphs from G(n, p) using cnp/ log np colors ? The answer to this question is affirmative. The algorithm consists mainly of finding disjoint independent sets, whose union covers the vertex set. For p large these independent sets can be found by a natural greedy algorithm, which works at follows [Bol] [FM] ALGORITHM 1. Suppose s independent sets have been found. Let Gs be the graph obtained by deleting all these independent sets. In Gs choose the vertex of smallest degree and delete its neighbors. Continue with the remaining graph until no vertex is left. The chosen vertices form the (s + 1)th independent set. By Theorem 1.2, it is obvious that the number of colors used is best possible, up to a constant factor. Now let us consider the general case of list-coloring hypergraphs. An (randomized) algorithm A is (H, t)-successful if, given a hypergraph H with any collection L of lists of size t, A can properly color H using L with probability at least 1/2. We say that A is almost surely t-successful if chosen H randomly from the distribution generalized by H(k, n, p), the probability that A is (H, t) successful is 1 − o(1). In other words, A is almost surely t-successful if it can color almost every hypergraph using any assignment of lists of size t. Theorem 4.1. For all ², δ > 0, there is a constant c such that if p satisfies n² < d(n) = pnk−1 /(k − 2)! < δn, then one can find a polynomial algorithm A which is almost surely cd(n)/ log d(n)-successful. Proof. The proof follows from the proof of Main Theorem and the fact that the proofs of Theorems 2.1-2.3 are algorithmic. The proof of Theorem 2.1 implies the following (see [Vu] for more detail) (i) If a graph G satisfies the conditions in 2.1 and np > n0.75 then there is a constant c and a deterministic algorithm which properly colors G, given any assignment of lists of size cnp/ log(np). A similar statement was proved in [AKS] with somewhat different parameters. The algorithm combines Algorithm 1 with the well known maximum matching algorithm [Lov], and is described below. ALGORITHM 2. The input is a graph G satisfying the conditions of Theorem 2.1 (for the set of parameters p, ², ²0 , γ determined in Case 1 of the proof of Main Theorem) and an arbitrary assignment of lists of size cnp/ log(np). The agorithm has two phases Generic steps. Order the set of colors arbitrarily. For each color c consider the set V (c) of those vertices, whose actual list contains c. If |V (c)| > (np)1−γ/2 , use Algorithm 1 to greedily choose an independent subset of V (c). Color the vertices of this set with c, and delete these vertices from the graph. Remove c from the lists of the remaining vertices and
12
VAN H. VU
move to the next generic step with the remaining graph (and lists). If |V (c)| ≤ (np)1−γ , consider the next color. If no color satisfies V (c) > (np)1−γ/2 , move to Final Step. Final Step. Let Gf be the remaining graph. Construct a bipartite graph B in the following way. The two color sets of B are Cf and Vf , where Cf is the set of all colors remaining in the lists of Gf and Vf is the set of vertices of Gf . Connect a color c with a vertex v if the list of v contains c. Find a matching which covers Vf , and color each vertex with the color determined by the matching. The main idea of the proof of Theorem 2.1 is to show that if the constant c in the input is sufficiently large, then at Final Step a matching covering Vf exists. Now we deal with the case the degrees in G are small (< n0.75 ). The proof of Theorem 2.3 implies the following: (ii) If a graph G satisfies the conditions in Theorem 2.3 and ∆ > n² , then for sufficiently large constant c, there is a randomized polynomial algorithm which can colors G with any list of size c∆/ log ∆ with probability close to 1. This algorithm is based on the so-called “nibble method”, following the approach of Johansson in [Joh]. Given a graph G1 with lists (not necessarily of the same cardinality) of colors assinged to its vertices, we call G1 greedily colorable if the minimum list cardinality is larger than the maximum degree of G1 . The algorithm mentioned in (ii) works roughly as follows. At each step choose a small random set of vertices. Color the vertices in this set if possible, and remove the already used colors to avoid conflict in future steps. Continue until the remaining uncolored graph is greedly colorable. In order to make this process work, the small set of vertices has to be chosen carefully from a properly defined space, and many side constrains also need to be maintained at each step. In the following we give a detailed description of the algorithm. ALGORITHM 3. The input is a graph G satisfying the condition in Theorem 2.3 (for α = 0.1), and an arbitrary assignment of lists of size c∆/ log ∆. Beginning State. For each list Lv and a color γ ∈ Lv , set p0v (γ) = 1/|Lv |. Let p0v = {p0v (γ)|γ ∈ Lv } and p0 = ∆−.99 . Set G0 = GH ,
Γ0 = {p0v }v ,
B 0 (v) = ∅ for every v,
V 0 = V (GH ).
Generic Steps. At a generic step i, we would work with V i , Γi etc. In order to avoid complicated notations, we will omit the superscript and put a prime on the outcome of the operation (V 0 stands for V i+1 , for instance). First check if the graph is greedily colorable. If yes, move to the Final Step. If no, execute the following two operations: Color. Consider Γ = ∪v pv . For each pv (γ) ∈ pv define a (0, 1) random variable tv (γ) with expected value θpv (γ). All random variables tv (γ) are to be independent. At a vertex v we say that a color γ with positive weight survives if γ ∈ / B(v) and tu (γ) = 0 for all u ∼ v. Color v with γ if γ survives and tv (γ) = 1 (if there are many such γ’s, use any of them). Reset. Now define p0v (γ) as follows: • If pv (γ)/E(I(γsurvives)) < p0 then let p0v (γ) = pv (γ)I(γ survives)/E(I(γsurvives)). • If pv (γ)/E(I(γsurvives)) ≥ p0 or γ ∈ B(v) we toss a biased coin and let p0v (γ) = p0 with probability ν = pv (γ)/p0 and p0v (γ) = 0 with probability 1 − ν.
ON THE CHOICE NUMBER OF RANDOM HYPERGRAPHS
13
Here I(A) is the indicator function of A; I(A) = 1 if A occurs and I(A) = 0 otherwise. Let C denote the set of colored vertices. Now define V 0 = V \C,
Γ0 = {p0v }v∈V 0 ,
B 0 = {γ|p0v (γ) = p0 },
G0 = G(V 0 ).
Final step. Color the remaining (greedily colorable) graph in (of course) greedy manner. Pick an arbitrary vertex v, color v by one of the colors of remaining in its list, say γ. Delete γ from all the lists of the neighbors of v and continue until every vertex is colored. The core of the proof of Theorem 2.3 is to show that with high probability, at some step we receive a greedily colorable graph. Now let us describe the algorithm which list-colors hypergraphs. We need to consider two cases: p (d(n) ) large and p (d(n)) small. Suppose d(n) > n0.75 . The algorithm works as follows. The input is a random hypergraph H and a collection of lists L. Accept H if GH satisfies the conditions of 2.1 (with the parameters determined in the proof of case 1 of Main Theorem) and reject H otherwise. By the proof of case 1 of Main Theorem, H is accepted with very high probability. If H is accepted, we apply Algorithm 2 to color GH using L; otherwise, do nothing. Algorithm 2 is deterministic, so once a hypergraph is accepted, it will be colored properly. Suppose n0.75 > d(n) > n² . Now accept H if GH satisfies the condition of 2.3 (with α = 0.1, as provided by Case 2 in the proof of Main Theorem), and apply Algorithm 3 to color GH if it is accepted. Again the proof of Main Theorem (case 2) implies that H is accepted with very high probability. The difference here is that Algorithm 3 is randomized, so once H is accepted (which guarantees that H is cd(n)/ log d(n) choosable), we only know that it will be properly colored with probability close to 1. ¤ §5 REMARKS AND OPEN QUESTIONS General remarks. • Let us consider a slightly more general of random hypergraphs. Fix a constant ¡ model ¢ k and for each k 0 ≤ k, choose each of kn0 subsets of size k 0 of V with probability pk0 . It results in a (non-uniform) hypergraph with edge cardinality at most k. Let d(n) = Pk k0 −1 0 /(k 0 − 2)!, then it is apparent that Main Theorem holds for this model of k0 =1 pk n random hypergraphs, assuming d(n) = O(n) (Theorem 1.3 was originally proved for this model). • The constant factor 0.9 in Theorem 1.5 and Main Theorem is ad hoc and can be replaced by any positive constant less than 1. In fact, the same statement holds even when one replaces 0.9 by 1 − f (n) with f (n) being a function tending slowly to 0 (see the proof of Theorem 2.1 in [Vu]). • It is proved in Main Theorem that when d(n) → ∞ the bounds on choice number fail with super-polynomially small probability. One may ask if the same holds when d(n) is a constant. It turns out to be not the case. Fix d(n) = ∆, and for the sake of simplicity, let 2 k = 2. The probability that there is a clique of size K = c2 ∆ is larger than n−K , where c2 is the constant in the upper bound of the Main Theorem. Consequently, with at least this probability the chromatic number is at least K > c2 ∆ > c2 d(n)/ log d(n). On the other 2 hand, n−K is not super-polynomially small. Brooks type bound The general Brooks conjecture says that if a graph has maximal degree ∆ and does not have a small clique, then χ(G) is considerably smaller than ∆. This conjecture is a major
14
VAN H. VU
conjecture in the theory of graph coloring and has been considered by various researchers (see [JT] for a survey). Recently, J.H. Kim [Kim] proved that if a graph has girth at least 5 and its maximal degree is ∆, then the (chromatic) choice number is at most c∆/ log ∆. With a very elegant and powerful proof, Johanson [Joh] strengthens this result by showing that the same bound holds for triangle free graphs. Theorem 5.1. (Johansson) If a graph G has maximal degree ∆ and contains no triangle, then there is a positive constant c such that: χl (G) < c∆/ log ∆. This bound is best possible up to a constant factor. Proven by applying Johansson’s method [Vu] , Theorem 2.3 further strengthens Theorem 5.1 by allowing the graph to have many triangles. More important, it suggests that the “triangle free” condition is not something “magical”, but rather a special case of a more general and natural phenomenon. A result similar to Theorem 2.3 can be shown for hypergraphs. A triangle in a (not necessarily uniform) hypergraph H is a triple of vertices x, y, z such that all three pairs (x, y), (y, z) and (z, x) are adjacent. Theorem 5.2. Suppose that the hypergraph H has maximal degree ∆ and each pair of adjacent vertices is contained in at most ∆1−α triangles, for some constant α. Then there is a constant c such that χl (H) ≤ c∆/ log ∆. Weak choice numbers In this paper we consider the strong proper coloring, which requires that the vertices in an edge must have different colors. Another popular definition of a proper coloring is the so-called weak coloring. In a proper weak coloring, every edge must have at least two colors, that is, no edge is monochromatic. For graphs, the notions of strong and weak coloring are the same; but for hypergraphs, they are quite different. Using the notion of weak proper coloring, we can define weak chromatic number and weak choice number accordingly. The asymptotic behavior of the weak chromatic number of random hypergraphs is determined in [KS]. In a forthcomming paper [KrV], Krivelevich and the present author compute the sharp order of magnitude of the weak choice number of random hypergraphs. It turns out that if a random hypergraph is sufficiently dense, then its weak chromatic number and weak choice number are asymptotically the same. Open questions. • It is proved that there are positive constants c1 and c2 such that c1 < χl (H(k, n, p))/χ(H(k, n, p)) < c2 almost surely. It remains an open problem to determine the best value of c1 and c2 . It is clear that one can set c1 = 1, and it seems plausible that c2 = 1 + o(1). However, our approach, in particular the methods used to prove Theorem 2.1-2.3 are not strong enough to prove this. Very recently, Krivelevich and Sudakov computed the sharp values of the constants c1 and c2 in Theorem 1.3 [KS]. It turned out that one can set c1 = 1/2 − o(1) and c2 = 1/2 + o(1), for all value of p such that d(n) = O(n). We conjecture that the
ON THE CHOICE NUMBER OF RANDOM HYPERGRAPHS
15
same constants can be used for Main Theorem. It seems that the method used in [KS] could be modified to prove the conjecture for the case the random hypergraph is dense (d(n) > nf (k) , for some f (k)). However when d(n) is small the conjecture appears to be hard. • The problem of determining the concentration of χ(G(n, p)) and χl (G(n, p)) obtains considerable attention. It was shown by L Ã uczak [ÃLuc2] that if p < n−5/6 then χ(G(n, p)) concentrates on only two values. This result was strengthened by Alon and Krivelevich, who proved that the same result holds for p < n−1/2−² , for any positive constant ². Using the approach in [ÃLuc2] and [AK], it was shown in [AKS] that if p < n−3/4−² then χl (G(n, p)) also concentrates on two values. We conjecture that if d(n) is a constant, then the situation is even better, i.e., χl (G(n, p)) (and more generally χl (H(k, n, p)) ) concentrates on only one value. We say that d is ²-good if there is a number r such that P r(χl H(k, n, p) = r) > 1 − ² if n is sufficiently large, where p = p(n) is chosen so that d(n) = pnk−1 /(k − 2)! = d. Conjecture 5.3. For any ² > 0, the set D = {d|d not ² -good} has zero Lebesgue measure. This conjecture is supported by the following result, proved by Achlioptas and Friedgut [AF]. Theorem 5.4. Let ² > 0 be some constant. For a given n, define a number x > 1 to be bad if the chromatic number of G(n, x/n) is not determined with probability greater than 1 − ², that is, ∀k, Pr[χ(G(n, x/n)) = k] ≤ 1 − ². For a given bound y > 1 define X²y = {x : y > x > 1, and x is bad} . Then for every ² > 0, and every y, lim m(X²y ) = 0
n→∞
where m is the Lebesgue measure. Acknowledgement. We would like to thank the referee for his careful reading and useful comments. REFERENCES [AF] D. Achlioptas and E. Friedgut, A sharp threshold for k-colorability, preprint. [Alo] N. Alon, Restricted colorings of graphs, Surveys in Combinatorics, Proc 14th British Combinatorial Conference, London Mathematical Society Lecture Notes Series 187, edited by K. Walker, Cambridge Univ. Press, 1993, 1-33. [AK] N. Alon and M. Krivelevich, On the concentration of the chromatic number of random graphs, Combinatorica 17 (1997), 401-426. [AKS] N. Alon, M. Krivelevich and Sudakov, List coloring of random and pseudo-random graphs, submitted. [AS] N. Alon and J. Spencer, The Probabilistic Method, Wiley, New York, 1992. [Bol] B. Bollob´as, Random Graphs, Academic Press, London, 1985. [Bol2] B. Bollob´as, The chromatic number of random graphs, Combinatorica 8 (1988), 49–55.
16
VAN H. VU
[ERT] P. Erd˝os, A. L. Rubin and H. Taylor, Choosability in graphs, Proc. West Coast Conf. on Combinatorics, Graph Theory and Computing, Congressus Numerantium XXVI, 1979, 125–157. [FM] A. Frieze and C.J.H. McDiarmid, Algorithmic theory of random graphs, Random structures and algorithms 10 (1997), 5–42. [GM] G. R. Grimmett and C.J.H. McDiarmid, On coloring random graphs, Proccedings of the Cambridge Philosophial Society 77 (1971) 313–324. [Joh] A. Johansson, Asymptotic choice number for triangle free graphs, DIMACS Technical Report 91–95 (1996). [JT] T.R. Jensen and B. Toft, Graph coloring problems , Wiley, 1995. [KL] M. Karonski and T. L Ã uczak, Random hypergraphs, Erd˝ os is eighty (Vol 2), 283-293. [Kim] J. H. Kim, On Brooks’ theorem for sparse graphs, Combinatorics, Probability and Computing 4 (1995), 97–132. [KV] J.H. Kim and V.H. Vu, Small complete arcs of finite projective planes, preprint. [KM] A. Kostochka and N. Mazurova, An inequality in the theory of graph coloring, Diskret. Analiz. 30 (1977), 23–29. [KrV] M. Krivelevich and V.H. Vu, The weak choice number of random hypergraphs, in preparation. [KS] M. Krivelevich and B. Sudakov, The chromatic number of random hypergraphs, Random structures and algorithms 12 (1998), 381-402. [Lov] L. Lov´ asz, Problem and Exercises in Combinatorics, North Holland 1979, chapter 7. [ÃLuc], T. L Ã uczak, The chromatic number of random graphs, Combinatorica 11 (1991), 45-54. [ÃLuc2] T. L Ã uczak, A note on the sharp concentration of the chromatic number of random graphs, Combinatorica 11 (1991), 295–297. [Mat] D.W. Matula, Expose-and-Merge exploration and the chromatic number of random graphs, Combinatorica 9 (1987), 275–284. [Sch] J. Schmidt, Probabilistic analysis of strong hypergraph coloring algorithm and the strong chromatic number, Discrete Math. 66 (1987), 259-277. [Sha] E. Shamir, Chromatic numbers of random hypergraphs and associated graphs, Advances in computing research 5 (1989) 127-142. [SSU] J. Schmidt-Pruzan, E. Shamir and E. Upfal, Random hypergraph coloring and the weak chromatic number, Journal of Graph Theory 9 (1985), 347-362. [Viz] V. G. Vizing, Coloring the vertices of a graph in prescribed colors (in Russian), Diskret. Analiz. No. 29, Metody Diskret. Anal. v. Teorii Kodov i Shem 101 (1976), 3–10. [Vu] Van H. Vu, On some simple degree conditions which guarantee the upper bound on the chromatic (choice) number of random graphs, submitted to Journal of Graph Theory.
Van H. Vu Department of Mathematics, Yale University 10 Hillhouse, New Haven, CT-06520 May 18, 1998
Abstract.
We generalize the notion of choice number from graphs to hypergraphs and estimate the sharp order of magnitude of the choice number of random hypergraphs. It turns out that the choice number and the chromatic number of a random hypergraph have the same order of magnitude, almost surely. Our result implies an earlier bound on the chromatic number of random hypergraphs, proved by Schmidt [Sch] using a different method.
§1 INTRODUCTION
¡ ¢ The k-uniform random hypergraph H(k, n, p) is obtained by choosing each of nk kelement subsets of a vertex set V (|V | = n) independently with probability p. The chosen subsets are the hyperedges of the hypergraph. Two vertices are adjacent if there is a hyperedge containing both of them. For k = 2, we have the usual definition of the random graph G(n, p) (see [Bol], for instance). In the whole paper, k is a fixed constant, and n tends to infinity. We say that H(k, n, p) has a property P almost surely, if the probability that H(k, n, p) satisfies P is 1 − o(1). Moreover, if the probability that P does not hold is of order O(exp(−ω(log n)), that is, super-polynomially small, then we say that H(k, n, p) has P with very high probability . This notion gives more accurate information about the rate of decay of the error term o(1), which could ¡n¢ be important in applications. We identify H(k, n, p) with the space S generated by k i.i.d (0, 1) random variables tA ’s with mean p, A ∈ A, where A is the family of all subsets of size k of V . The points of this space correspond to the k-uniform hypergraphs on V , and the probability that H(k, n, p) has a property P is equal to the measure of the subset (of S) consisting of those points (hypergraphs) which satisfy P . In this paper, we make use of standard asymptotic notations such as o, O, ω, Ω and Θ. Readers unfamiliar to these notations can check [Bol2] or [AS] for definitions. Given a graph G, a proper coloring of G is a coloring such that no two adjacent vertices receive the same color. The chromatic number of G, χ(G), is the smallest number of colors one needs to use to color G properly. This notion could be generalized for hypergraphs in many ways. In this paper, we will concentrate on the following definition, which generalizes the above notion of proper coloring in, perhaps, the most natural manner. Definition 1.1 A hypergraph H is properly colored (in strong sense) if no two adjacent vertices receive the same color. The strong chromatic number of H, χ(H), is the smallest number of colors one needs to use to properly color H.
2
VAN H. VU
The problem of determining the chromatic number χ(G(n, p)) of the random graph G(n, p) is very well known and was considered by several researchers. Results of Bollob´as [Bol] [Bol2], L Ã uczak [ÃLuc], Matula [Mat], Grimmett and McDiarmid [GM] show that Theorem 1.2. There are positive constants c1 and c2 such that almost surely c1 np/ log(np) < χ(G(n, p)) < c2 np/ log(np). The sharp values of the constants c1 and c2 were determined by Bollob´as [Bol2] for large p, and by L Ã uczak [ÃLuc] for small p, so the problem is completely solved. To generalize Theorem 1.2, various authors studied the chromatic number of random hypergraphs [Sch], [SSU], [Sha]. Althought the above notion of (strong) chromatic number of hypergraphs is a natural generalization of that of graphs, the problem is more complex and requires new ideas. The sharp order of magnitude of χ(H(k, n, p)) was found by Schmidt [Sch] Theorem 1.3. Suppose p = p(n) is such a function of n that d(n) = nk−1 p/(k − 2)! tends to infinity but d(n) = O(n). Then there exists positive constants c1 and c2 such that almost surely c1 d(n)/ log(d(n)) < χ(H(k, n, p)) < c2 d(n)/ log(d(n)) i.e, χ(H(k, n, p)) = Θ(d(n)/ log(d(n))). Theorem 1.3 generalizes Theorem 1.2 (if k = 2 then d(n) = np); however, in [Sch] it was not clear what the sharp values of c1 and c2 are , when k > 2. The computation in [Sch] showed that one can set c1 ∼ 1/2 and c2 ∼ 2. In [Sha], Shamir proved that if d(n) > n6/7 , then c1 ∼ c2 ∼ 1/2. This result was extended to all values of d(n) in a recent paper of Krivelevich and Sudakov [KS]. To describe our goal, we now introduce the notion of choice number of hypergraphs. The notion of choice number of graphs was introduced by Erd˝os, Rubin and Taylor in [ERT], and also by Vizing in [Viz]. Assign to each vertex v in a graph G a list Lv of t colors (different vertices may have different lists), a list coloring is a coloring in which every vertex is colored by a color from its own list. The choice number χl (G) of the graph is the least number t such that there exists a proper list coloring for every assignment of lists of size t to the vertices. It is fairly trivial that χl (G) ≥ χ(G) for all G. Recently, choice number becomes an exciting research topics, leading to many beautiful questions and theorems (see [Alo] for a survey). Following Definition 1.2, we now define the (strong) choice number of a hypergraph. Definition 1.4. Given a hypergraph H, the choice number of H, χl (H), is the smallest t such that H admits a proper coloring for every possible assignment of lists of size t. The huge number of possible assignments makes the problem of estimating choice numbers much more complex than that of chromatic numbers. It is already hard to compute the choice number in relatively simple graphs, for which the chromatic numbers are easily seen. Let us, for instance, consider the complete bipartite graph Kn,n . While its chromatic number is 2, we do not know the exact value of its choice number. Furthermore, it was proven by Erd˝os, Rubin and Taylor that χl (Kn,n ) = (1 + o(1)) log n [ERT]. This shows that the ratio χl (G)/χ(G) (which is lower bounded by 1) can be arbitrarily large. As the
ON THE CHOICE NUMBER OF RANDOM HYPERGRAPHS
3
worst case analysis is concerned, it is shown in [ERT] that computing the choice number of a given graph is strictly harder than computing the chromatic number, under a commonly believed hypothesis that coN P 6= N P . Althought in many special graphs, the choice number could be much larger than the chromatic number, it turns out that for a typical graph, they have the same order of magnitude. The present author [Vu] showed that almost surely χl (G(n, p) = Θ(c(²)np/ log(np)) for 0.9 > p > n−1 log1+² n with any positive constant ². In [AKS], Alon, Krivelevich and Sudakov showed that Θ(np/ log(np)) is the right order of magnitude of χl (G(n, p)), for all p such that 2 ≤ np ≤ n/2. Together, we have Theorem 1.5. There are positive constants c1 and c2 such that for all p, 2 ≤ np ≤ 0.9n, almost surely c1 np/ log(np) < χl (G(n, p)) < c2 np/ log(np). In this paper, we determine the sharp order of magnitude of the choice number of random hypergraphs. Our main theorem (generalizing Theorem 1.5) shows that almost surely, the choice number and the chromatic number of a random (k-uniform) hypergraph have the same order of magnitude (the order of magnitude of χ(H(k, n, p)) was shown in Theorem 1.3). Main Theorem. Assume that k > 2, there are positive constants c1 and c2 such that c1 d(n)/ log d(n) < χl (H(k, n, p)) < c2 d(n)/ log d(n) almost surely, where d(n) = nk−1 p/(k − 2)! and d(n) = O(n). Moreover, when d(n) = ω(1), the above statement is true with very high probability. If k = 2, the same result holds under an additional assumption that d(n) < 0.9n. The Main Theorem implies Schmidt’s theorem (and therefore it also implies Theorem 1.2), with somewhat more generous constants. On the other hand, it gives a stronger statement (compared to Theorems 1.3 and 1.5) in the sense that it shows if d(n) tends to infinity then the probability that the choice number has wrong order of magnitude not only tends to 0, but does it super-polynomially fast. This guarantees that if one randomly picks polynomially many hypergraphs, then with very high probability their (strong) choice numbers (or chromatic numbers) differ by only a constant factor, given d(n) → ∞. Such property might be useful for applications in computer science, especially in the area of randomized algorithms, where one often has to make (usually polynomially many) random samples from some space. In the next section, we discuss the main difficulties concerning choice number and random hypergraphs, and present the main idea of the proof and some key tools. The proof of Main Theorem appears in Section 3. In Section 4, we discuss the algorithmic aspect of Main Theorem. We show that for most values of p, there is a polynomial time (randomized) algorithm, by which one can properly color almost every hypergraph in H(k, n, p) using any assignment of lists of size c2 d(n)/ log d(n). We conclude with Section 5, which contains few remarks and open questions. §2 PREMILINARIES A set of vertices is independent if its vertices are pairwise non-adjacent. Given a hypergraph H, we denote by α(H) the cardinality of the largest independent set of H. It is clear that χ(H) ≥ n/α(H).
4
VAN H. VU
The lower bound in Theorem 1.3 (and Theorem 1.2) is proved by a simple “first moment” argument, which shows that almost surely, H(k, n, p) cannot have a “big” independent set with more than cn log(d(n))/d(n) vertices, for some constant c (see [Bol], [Sch], for instance). It is easily seen that if d(n) = ω(1), then for appropriate c, the probability of having a “big” independent set is super-polynomially small. Since the choice number is at least the chromatic number, this implies the lower bound in Main Theorem. The more involved part of the proof of Main Theorem is to show the upper bound. Let us first discuss the difficulties one may face when one needs to estimate the choice number instead of the chromatic number. The general idea in the proofs of the upper bounds for chromatic numbers in Theorems 1.3 and 1.2 is to show that almost surely there are many disjoint independent sets of proper order (Θ(n log d(n)/d(n))), whose union covers almost every vertex. The rest (and negligible) part of the vertex set can be colored greedly afterwards. Unfortunately, this technique does not always help to give an upper bound for the choice number. Even when the graph can be covered by only two independent sets (as shown in the example of Kn,n ), the choice number could be arbitrarily large. Therefore, we need to find a different approach to tackle the problem. Before showing the idea, let us point out another potential obstruction. Consider a hypergraph H on a set V of n elements. Let GH denote the graph on V obtained by joining all pairs (x, y), where x and y are adjacent in H. Following the definition of choice number of hypergraphs, it is easy to see that χl (H) = χl (GH ). Readers may think that by this simple trick, the problem on hypergraphs is reduced to graphs, and one could apply Theorem 1.5 to prove Main Theorem. However, the situation is not so simple; while H(k, n, p) is a random hypergraph, it is not true that GH(k,n,p) is a random graph. The fact that GH(k,n,p) is not random turns out to be a serious obstruction in the investigation of many properties and makes the study of random hypergraphs considerably more involved than that of random graphs (see [KL] for a collection of important, unsolved problems concerning random hypergraphs). Let us, for instance, consider the problem of computing α(H(k, n, p)) = α(GH(k,n,p) ). For k = 2 (GH(k,n,p) = G(n, p)), it is quite easy to compute α(G(n, p)) up to a 1 + o(1) multiplicative factor [Bol] [AS], while for k > 2 such result was obtained just very recently [KS]. The reason is that when k = 2, the property that a subset V 0 of GH(k,n,p) is independent is determined by the edges contained in V 0 . So if V 0 and V ” are disjoint, then the events that V 0 and V ” are independent sets are probabilistically independent, which makes the computation easy. On the other hand, if k > 2, the property that V 0 is independent is determined by the hyperedges intersecting V 0 in at least 2 vertices, and the above mentioned feature is lost. The key idea, which helps us to handle these obstructions is the following observation, pointed out by the present author in [Vu]: The desired upper bound on the choice number of a graph can be derived from some local properties. Let us describe what we mean by this observation. Consider two vertices x and y, the degree of x is the number of vertices adjacent to x, while the codegree of x and y is the number of vertices adjacent to both x and y. In a random graph G(n, p), the expected value of a degree is roughly np, and that of a codegree is roughly np2 . It is shown that if in a deterministic graph G, the degrees and codegrees are “not too far” from these values (np and np2 , resp), then the choice number of G is O(np/ log(np)), which is the bound we desire. This observation, at one strike, helps us to solve both of the above mentioned difficulties.
ON THE CHOICE NUMBER OF RANDOM HYPERGRAPHS
5
First, it gives a new approach to prove the desired bound on the choice number. Moreover, since the condition involves only degrees and codegrees, it is relatively easy to check, even in GH(k,n,p) , which is not a random graph in the traditional sense. The proof of Main Theorem consists essentially of the verification of the degree-codegree condition, which varies depending on the order of magnitude of p. Let us now present few theorems to support the above observation. In Section 4, we will sketch the proofs. For complete proofs, we refer to [Vu]. Theorem 2.1. Let γ be a positive constant less than 1/2. Assume that p, ² and ²0 satisfy 2γ−1
0.9 > p = ω(n 3−2γ ) (np)γ p−1 (² + ²0 ) = o(1). Consider a graph G on n vertices. Assume that every degree in G is between np(1 + ²) and np(1 − ²), and every codegree is between np2 (1 + ²0 ) and np2 (1 − ²0 ), respectively. Then χl (G) ≤ c
np γ log(np)
for some absolute constant c. Theorem 2.2. Assume that n−1 < p < n−² for some positive constant ². If G is a graph on n vertices with every degree at most anp and every codegree at most bnp2 , for some constants a and b, then there is a constant c such that χl (G) ≤ cnp/ log(np). Theorem 2.2 could be strengthened in the following way: first one can relax the condition on codegrees, allowing a larger multiplicative error term. Secondly, instead of requiring the codegree condition for every pair x and y, it is enough to keep this condition on those x and y where x and y are adjacent. To make it easier to apply, in the following Theorem we replace np by ∆. Theorem 2.3. Let α be a positive constant less than 1. Assume G is a graph on n vertices with maximal degree at most ∆, and every edge of G is contained in at most ∆1−α triangles. Then χl (G) ≤ c
∆ log(∆)
for some constant c. Remark. In [AKS], a theorem similar to Theorem 2.1 is proven (Theorem 1.2 in [AKS]). This theorem gives the same bound for the choice number under weaker conditions (it needs only one-side bounds on the degrees and codegrees), but for a smaller range of p (it is valid 1 for p > n−1/12 , while Theorem 2.1 is valid for p > n− 3 +² for any ² > 0). We will also need the following technical Lemmas. The first Lemma is a well known Theorem due to Azuma, which is very useful in proving strong concentration of functions satisfying the Lipchitz condition.
6
VAN H. VU
Lemma 2.4. (Azuma theorem) Let 0 = X1 , . . . , Xm be a martingale with |Xi+1 − Xi | ≤ 1 for all 0 ≤ i < m. Then for any λ > 0, 2
P r(Xm > λm1/2 ) < e−λ
/2
.
The martingale we consider in this paper is the so-called edge exposure martingale (for the general set up we refer to [AS]). Lemma 2.4 implies (with room to spare) that if F is a (hyper)-graphtheoretic function, and changing one (hyper)-edge changes the value of F by at most a constant, then with probability 1 − exp(−Ω(log2 m)), F has value between E(F ) ± E(F )1/2 log m. In the proof, we will also need to show strong concentration of certain multi-variable polynomials, which do not satisfy the Lipchitz condition (changing one variable may change the value of the function significantly). The key tool to handle these cases is Lemma 2.5, which is a special case of a more general Lemma in [KV]. To state Lemma 2.5, we need some preparation. Let K be a r-uniform hypergraph with V (K) = {1, 2, ..., n}. Suppose ti , i = 1, , 2..., n are i.i.d. (0, 1) random variables and E(ti ) = p. Consider the following polynomial X Y
YK =
ts
e∈E(K) s∈e
where E(K) is the edge set of K. For each subset A of V (K), |A| ≤ r, define a hypergraph KA as follows • V (KA ) = V (K)\A, • E(KA ) = {B ⊂ V (KA ), B ∪ A ∈ E(K)}. Now let Ei (K) = maxA⊂V (K),|A|=i E(YKA ). It is clear that E0 (K) = E(YK ). Furthermore, let E = maxi≥0 Ei and E 0 = maxi≥1 Ei . Lemma 2.5. (Hypergraph Lemma) There exist positive numbers cr , dr depending only on r so that for any λ > 0 P r(|YK − E0 (K)| > cr (EE 0 )1/2 λr ) < dr exp(−λ + (r − 1) log n). In particular, when r is small (r ≤ 5, say) P r(|YK − E0 (K)| > (EE 0 )1/2 logr+1 n) = O(exp(−ω(log n))). Lemma 2.5 and its more general version in [KV] have many interesting applications, especially in situations where Azuma theorem cannot be applied, because of the failure of the Lipchitz condition. These applications will be discussed in another paper. The next two Lemmas reveal some useful information about GH(k,n,p) . In all proofs in the next Section and the remaining of this Section, we assume that d(n) = pnk−1 /(k − 2)! is sufficiently large, whenever needed. Lemma 2.6. Almost surely (1) For every X ⊂ V , |X| ≤ 2n/log 2 d(n) the induced subgraph of GH(k,n,p) spanned by X has less than (d(n)/ log d(n))|X| edges.
ON THE CHOICE NUMBER OF RANDOM HYPERGRAPHS
7
(2) All but at most n/ log2 d(n) vertices of G have degree less than 10kd(n). Moreover, if d(n) = ω(1), both statements hold with very high probability. Proof. This Lemma was proved in [AKS] for the special case k = 2, i.e., GH(k,n,p) is the random graph G(n, p). Although for k > 2, GH(k,n,p) is not random, somewhat similar computation could still be carried out. Consider a set X, denote by F (X) the number of hyperedges of H(k, n, p) which intersects X in at least two points. It is clear that k 2 F (X) is larger than the number of edges of the subgraph spanned by X (in GH(k,n,p) ). Similarly, let G(X) denote the number of hyperedges which intersects X in exactly one points. Set r = d(n)/k 2 log(d(n)), s = 2n/ log2 d(n), and assume |X| = i ≤ s. Without loss of generality we may assume that r and s are positive integers. We have µ¡ i ¢¡ n ¢¶ P r(F (X) > ri) < 2 k−2 pri ri ≤(
ei2 nk−2 ri ri eid(n) ri eik 2 log d(n) ri ) p (10ks0 d(n) − s0 d(n))/k > 9s0 d(n) = t. Therefore, X is a special set. Since the probability that a special set exists is o(1), we complete the proof of part (2) of the Lemma. To prove the “with very high probability” part of (1), it suffices to notice that if d(n) = ω(1), the last sum in (2.1) is super-polynomially small in n. Indeed, it is easy to show that the terms of this sum are decreasing, and the first term is super-polynomially small. Since there are less than n terms, we are done. The “with very high probability” part of (2) follows from the fact that (1) holds with very high probability (given d(n) = ω(1)) and the rightmost formula in (2.2) is super-polynomially small. ¤
8
VAN H. VU
Lemma 2.7. Assume that d(n) < log4 n, then with very high probability, there is a set X of o(log17 n) points such that in the subgraph of GH(k,n,p) spanned by V \X, each edge is contained in at most k triangles. Proof. Let A, B, C be three edges of the random hypergraph. We say that the triple {A, B, C} is bad if there are three different points x, y, z such that x ∈ A ∩ B, y ∈ A ∩ C and z ∈ B ∩ C. We write {x, y, z} ◦ {A, B, C} if x, y, z and A, B, C satisfy the above conditions. Furthermore, we say that the pair {A, B} is bad if |A ∩ B| ≥ 2. First we show that with very high probability the numbers of bad pairs and bad triples are small. It is easily checked that the number of bad pairs (as a random variable) is at most X
X
tA tB = Y.
x,y∈V {x,y}∈A∩B x6=y A6=B
Let us first estimate the expected value of Y µ ¶ µ ¶ n n E(Y ) ∼ (p )2 < d(n)2 ≤ log8 n. 2 k−2 It follows immediately from Markov inequality that P r(Y > log11 n) = O(1/ log3 n) = o(1), that is, almost surely there are at most log11 n bad pairs. However, we want to prove a stronger statement that the same event holds with very high probability, i.e, P r(Y > log11 n) is not only o(1), but super-polynomially small. Azuma theorem cannot be applied for this purpose, since Y has very large Lipchitzian coefficient (at ¡certain ¢ points, n−2 changing the value of one variable may change the value of Y as much as k−2 ). A very convenient tool for this situation is the Hypergraph Lemma (Lemma 2.5). Notice that the polynomial Y has degree r = 2. Moreover, E0 (Y ) = E(Y ) < log8 n and E2 (Y ) = O(1) and µ ¶ µ ¶ k n E1 (Y ) ≤ p = O(d(n)/n) = o(1). 2 k−2 So Hypergraph Lemma implies that with very high probability Y ≤ log11 n,i.e, with very high probability there are less than log11 n bad pairs. Now let us consider the number of bad triples. This number is upper bounded by X
X
x,y,z∈V
A,B,C∈A {x,y,z}◦{A,B.C}
tA tB tC = Y 0 .
In the summation we assume that x, y, z ( A, B, C) are different points (subsets), resp. In order to use Hypergraph Lemma, let us notice that E3 (Y 0 ) = O(1). Moreover, µ ¶ µ ¶ n n E(Y ) = E0 (Y ) ∼ (p )3 < d(n)3 ≤ log12 n, 3 k−2 0
0
µ 0
2
E1 (Y ) = O(1)p
¶µ
n k−1
¶
n k−2
= o(1),
ON THE CHOICE NUMBER OF RANDOM HYPERGRAPHS
9
¶ µ n E2 (Y ) = O(1)p = o(1). k−2 0
Since the polynomial has degree 3, Hypergraph Lemma thus implies that with very high probability Y 0 < log16 n, that is, the number of bad triples is less than log16 n with very high probability. Let X be the set of vertices covered by a bad pair or a bad triple, then |X| = o(log17 n). Setting W = V \X, one could prove without any difficulty that the induced subgraph spanned by W satisfies the statement of the Lemma. ¤ The last Lemma is deterministic, and was used in [AKS]. A similar statement was proven earlier in [ÃLuc]. Lemma 2.8. Let G = (V, E) be a graph on n vertices such that there is a subset U0 ∈ V of size (1 + o(1))n/ log2 d(n) such that the induced subgraph spanned by V \U0 has choice number of order O(d(n)/ log d(n)). Suppose further that every q < 2n/ log2 d(n) vertices of G spanned less than qd(n)/ log d(n) edges. Then χl (G) = O(d(n)/ log d(n)). §3 THE PROOF OF MAIN THEOREM Case 1: d(n) ≥ n0.9 . This case can be solved by applying Theorem 2.1 (with suitable parameters). To do this, we first need to estimate the degrees and codegrees of GH(k,n,p) . Consider a vertex n−2 x. For any y 6= x, the probability that x and y are not adjacent is (1 − p)(k−2) = q . Let 1
p1 = 1 − q1 , the expected value of number of neighbors of x is p1 n. It is easy to check that p1 n = Θ(d(n)), so we may assume that p1 > n−0.11 . Observe that dx is a random variable depending on the atom variables tA ’s, and changing the value of one of the tA ’s will change the value of dx by at most k − 1. So, by Azuma theorem, we have with very high probability that p1 n − n0.6 < p1 n − log2 n(p1 n)1/2 < dx < p1 n + log2 n(p1 n)1/2 < p1 n + n0.6 . It follows that with very high probability, all degrees in GH(k,n,p) are between p1 n − n and p1 n + n0.6 . Now consider the codegree dx,y of x and y. First we need to estimate the expected value of dx,y . Let M be expected value of the number ¡ ¢ of paths of length 3 in GH(k,n,p) ; by symmetry the expected value of dx,y equals M/ n2 . On the other hand, with very high probability, each vertex in GH(k,n,p) has degree between d1 = p1 n − n0.6 and d2 = p1 n + n0.6 , so¡ with very ¢ ¡ ¢high probability, the number of paths of length 3 in GH(k,n,p) is between n d21 and n d22 . This implies 0.6
n((p1 n)2 /2 − 2n1.6 ) < M < n((p1 n)2 /2 + 2n1.6 ). So p21 n − 4n0.6 < E(dx,y ) < p21 n + 4n0.6 . Using Azuma theorem (Lemma 2.4), one can show that with very high probability, p21 n − 5n0.6 < dx,y < p21 n + 5n0.6 . It follows that with very high probability, all codegrees of GH(k,n,p) are between p21 n − 5n0.6 and p21 n + 5n0.6 . To apply Theorem 2.1, set ² = ²0 = n−0.15 , γ = 0.01. Notice that since p1 > n−0.11 , the degrees and codegrees of GH(k,n,p) satisfy the conditions of Theorem 2.1 (with p1 playing the role of p), with very high probability. Therefore, by Theorem 2.1, we
10
VAN H. VU
have with very high probability χl (GH(k,n,p) ) = O(np1 / log(np1 )) = O(d(n)/ log d(n)), completing the proof. Case 2: log4 n < d(n) < n0.9 . Consider the graph GH(k,n,p) . Let us first estimate the degrees in GH(k,n,p) . For a point x ∈ V , let dx denote the degree of x in GH(k,n,p) , we have
(3.1)
dx ≤ (k − 1)
X
tA = Y.
x∈A,|A|=k
¡ n ¢ The expected value of Y is (k − 1)p k−1 ∼ d(n). Since this is a sum of independent variables, it is easy to show (via Chernoff bound) that with very high probability Y ≤ 2d(n) (here we need the assumption that d(n) > log4 n; if d(n) is too small, d(n) = log log n, for instance, the statement is not true). Consider a pair of vertices (x, y). Similarly to (3.1), we have
(3.2)
dx,y ≤
X
(
X
z∈V \{x,y} x,z∈A,|A|=k
tA )(
X
tB ) = Y 0 .
y,z∈B,|B|=k
¡ n ¢2 The expected value E(Y 0 ) = E0 (Y 0 ) of Y 0 is roughly n(p k−2 ) ∼ d(n)2 /n. To use ¡ ¢ n Hypergraph Lemma, notice that E1 (Y 0 ) = O(p k−2 ) = o(1) and E2 (Y 0 ) = O(1). Hypergraph Lemma yields that with very high probability Y < c(E(Y 0 ) + 1) log3 n, for some constant c. It is easy to check that c(E(Y 0 ) + 1) log3 n = o(d(n)0.9 ) for all d(n) in the interval of interest. ¡ ¢ Since there are only n2 pairs (x, y), we conclude that with very high probability every degree in GH(k,n,p) is at most 2d(n), and every codegree in GH(k,n,p) is less than d(n)0.9 . Set ∆ = 2d(n). It follows directly from Theorem 2.3 that there is a constant c such that with very high probability χl (GH(k,n,p) ) ≤ c∆/ log ∆ = cd(n)/ log d(n), which completes the proof for this case. Case (3): d(n) ≤ log4 n. One cannot repeat the argument used in the previous case in this case, because when d(n) is too small, the strong concentration results do not hold. The trick here is to “divide and conquer”, which was applied earlier for random graphs in [AKS]. Let U be the subset of V which contains all those vertices, which either have degree at least 10kd(n) or are covered by a bad pair or bad triple. Due to Lemma 2.7 and the second part of Lemma 2.6 , |U | < n/ log2 d(n) + log17 n = (1 + o(1))n/ log2 d(n) almost surely. Let W = V \U , and G0 the subgraph spanned by W . By the definition of U , the maximum degree of G0 is at most 10kd(n) almost surely. Moreover, by Lemma 2.7, almost surely, each edge in G0 is contained in at most k triangles. Without loss of generality, one can assume that d(n) > k. Set ∆ = 10kd(n), then the condition of Theorem 2.3 is satisfied by G0 almost surely. Therefore, by Theorem 2.3, there is a constant c such that the choice number of G0 is at most c∆/ log ∆ = cd(n)/ log d(n) almost surely Applying Lemma 2.8, we can conclude that almost surely χl (GH(k,n,p ) ≤ cd(n)/ log d(n), for some constant c.
ON THE CHOICE NUMBER OF RANDOM HYPERGRAPHS
11
To show that the bound holds with very high probability (given d(n) = ω(1)), it suffices to observe that under this assumption the statements in Lemma 2.6 hold with very high probability. ¤ §4 THE ALGORITHMIC ASPECT In this Section, we study the algorithmic aspect of Main Theorem. Let us first start with the special case of random graph and chromatic number. Theorem 1.2 states that the random graph G(n, p) has chromatic number at most cnp/ log np almost surely, for some constant c. We consider the following question Is there a polynomial algorithm which could color almost every graphs from G(n, p) using cnp/ log np colors ? The answer to this question is affirmative. The algorithm consists mainly of finding disjoint independent sets, whose union covers the vertex set. For p large these independent sets can be found by a natural greedy algorithm, which works at follows [Bol] [FM] ALGORITHM 1. Suppose s independent sets have been found. Let Gs be the graph obtained by deleting all these independent sets. In Gs choose the vertex of smallest degree and delete its neighbors. Continue with the remaining graph until no vertex is left. The chosen vertices form the (s + 1)th independent set. By Theorem 1.2, it is obvious that the number of colors used is best possible, up to a constant factor. Now let us consider the general case of list-coloring hypergraphs. An (randomized) algorithm A is (H, t)-successful if, given a hypergraph H with any collection L of lists of size t, A can properly color H using L with probability at least 1/2. We say that A is almost surely t-successful if chosen H randomly from the distribution generalized by H(k, n, p), the probability that A is (H, t) successful is 1 − o(1). In other words, A is almost surely t-successful if it can color almost every hypergraph using any assignment of lists of size t. Theorem 4.1. For all ², δ > 0, there is a constant c such that if p satisfies n² < d(n) = pnk−1 /(k − 2)! < δn, then one can find a polynomial algorithm A which is almost surely cd(n)/ log d(n)-successful. Proof. The proof follows from the proof of Main Theorem and the fact that the proofs of Theorems 2.1-2.3 are algorithmic. The proof of Theorem 2.1 implies the following (see [Vu] for more detail) (i) If a graph G satisfies the conditions in 2.1 and np > n0.75 then there is a constant c and a deterministic algorithm which properly colors G, given any assignment of lists of size cnp/ log(np). A similar statement was proved in [AKS] with somewhat different parameters. The algorithm combines Algorithm 1 with the well known maximum matching algorithm [Lov], and is described below. ALGORITHM 2. The input is a graph G satisfying the conditions of Theorem 2.1 (for the set of parameters p, ², ²0 , γ determined in Case 1 of the proof of Main Theorem) and an arbitrary assignment of lists of size cnp/ log(np). The agorithm has two phases Generic steps. Order the set of colors arbitrarily. For each color c consider the set V (c) of those vertices, whose actual list contains c. If |V (c)| > (np)1−γ/2 , use Algorithm 1 to greedily choose an independent subset of V (c). Color the vertices of this set with c, and delete these vertices from the graph. Remove c from the lists of the remaining vertices and
12
VAN H. VU
move to the next generic step with the remaining graph (and lists). If |V (c)| ≤ (np)1−γ , consider the next color. If no color satisfies V (c) > (np)1−γ/2 , move to Final Step. Final Step. Let Gf be the remaining graph. Construct a bipartite graph B in the following way. The two color sets of B are Cf and Vf , where Cf is the set of all colors remaining in the lists of Gf and Vf is the set of vertices of Gf . Connect a color c with a vertex v if the list of v contains c. Find a matching which covers Vf , and color each vertex with the color determined by the matching. The main idea of the proof of Theorem 2.1 is to show that if the constant c in the input is sufficiently large, then at Final Step a matching covering Vf exists. Now we deal with the case the degrees in G are small (< n0.75 ). The proof of Theorem 2.3 implies the following: (ii) If a graph G satisfies the conditions in Theorem 2.3 and ∆ > n² , then for sufficiently large constant c, there is a randomized polynomial algorithm which can colors G with any list of size c∆/ log ∆ with probability close to 1. This algorithm is based on the so-called “nibble method”, following the approach of Johansson in [Joh]. Given a graph G1 with lists (not necessarily of the same cardinality) of colors assinged to its vertices, we call G1 greedily colorable if the minimum list cardinality is larger than the maximum degree of G1 . The algorithm mentioned in (ii) works roughly as follows. At each step choose a small random set of vertices. Color the vertices in this set if possible, and remove the already used colors to avoid conflict in future steps. Continue until the remaining uncolored graph is greedly colorable. In order to make this process work, the small set of vertices has to be chosen carefully from a properly defined space, and many side constrains also need to be maintained at each step. In the following we give a detailed description of the algorithm. ALGORITHM 3. The input is a graph G satisfying the condition in Theorem 2.3 (for α = 0.1), and an arbitrary assignment of lists of size c∆/ log ∆. Beginning State. For each list Lv and a color γ ∈ Lv , set p0v (γ) = 1/|Lv |. Let p0v = {p0v (γ)|γ ∈ Lv } and p0 = ∆−.99 . Set G0 = GH ,
Γ0 = {p0v }v ,
B 0 (v) = ∅ for every v,
V 0 = V (GH ).
Generic Steps. At a generic step i, we would work with V i , Γi etc. In order to avoid complicated notations, we will omit the superscript and put a prime on the outcome of the operation (V 0 stands for V i+1 , for instance). First check if the graph is greedily colorable. If yes, move to the Final Step. If no, execute the following two operations: Color. Consider Γ = ∪v pv . For each pv (γ) ∈ pv define a (0, 1) random variable tv (γ) with expected value θpv (γ). All random variables tv (γ) are to be independent. At a vertex v we say that a color γ with positive weight survives if γ ∈ / B(v) and tu (γ) = 0 for all u ∼ v. Color v with γ if γ survives and tv (γ) = 1 (if there are many such γ’s, use any of them). Reset. Now define p0v (γ) as follows: • If pv (γ)/E(I(γsurvives)) < p0 then let p0v (γ) = pv (γ)I(γ survives)/E(I(γsurvives)). • If pv (γ)/E(I(γsurvives)) ≥ p0 or γ ∈ B(v) we toss a biased coin and let p0v (γ) = p0 with probability ν = pv (γ)/p0 and p0v (γ) = 0 with probability 1 − ν.
ON THE CHOICE NUMBER OF RANDOM HYPERGRAPHS
13
Here I(A) is the indicator function of A; I(A) = 1 if A occurs and I(A) = 0 otherwise. Let C denote the set of colored vertices. Now define V 0 = V \C,
Γ0 = {p0v }v∈V 0 ,
B 0 = {γ|p0v (γ) = p0 },
G0 = G(V 0 ).
Final step. Color the remaining (greedily colorable) graph in (of course) greedy manner. Pick an arbitrary vertex v, color v by one of the colors of remaining in its list, say γ. Delete γ from all the lists of the neighbors of v and continue until every vertex is colored. The core of the proof of Theorem 2.3 is to show that with high probability, at some step we receive a greedily colorable graph. Now let us describe the algorithm which list-colors hypergraphs. We need to consider two cases: p (d(n) ) large and p (d(n)) small. Suppose d(n) > n0.75 . The algorithm works as follows. The input is a random hypergraph H and a collection of lists L. Accept H if GH satisfies the conditions of 2.1 (with the parameters determined in the proof of case 1 of Main Theorem) and reject H otherwise. By the proof of case 1 of Main Theorem, H is accepted with very high probability. If H is accepted, we apply Algorithm 2 to color GH using L; otherwise, do nothing. Algorithm 2 is deterministic, so once a hypergraph is accepted, it will be colored properly. Suppose n0.75 > d(n) > n² . Now accept H if GH satisfies the condition of 2.3 (with α = 0.1, as provided by Case 2 in the proof of Main Theorem), and apply Algorithm 3 to color GH if it is accepted. Again the proof of Main Theorem (case 2) implies that H is accepted with very high probability. The difference here is that Algorithm 3 is randomized, so once H is accepted (which guarantees that H is cd(n)/ log d(n) choosable), we only know that it will be properly colored with probability close to 1. ¤ §5 REMARKS AND OPEN QUESTIONS General remarks. • Let us consider a slightly more general of random hypergraphs. Fix a constant ¡ model ¢ k and for each k 0 ≤ k, choose each of kn0 subsets of size k 0 of V with probability pk0 . It results in a (non-uniform) hypergraph with edge cardinality at most k. Let d(n) = Pk k0 −1 0 /(k 0 − 2)!, then it is apparent that Main Theorem holds for this model of k0 =1 pk n random hypergraphs, assuming d(n) = O(n) (Theorem 1.3 was originally proved for this model). • The constant factor 0.9 in Theorem 1.5 and Main Theorem is ad hoc and can be replaced by any positive constant less than 1. In fact, the same statement holds even when one replaces 0.9 by 1 − f (n) with f (n) being a function tending slowly to 0 (see the proof of Theorem 2.1 in [Vu]). • It is proved in Main Theorem that when d(n) → ∞ the bounds on choice number fail with super-polynomially small probability. One may ask if the same holds when d(n) is a constant. It turns out to be not the case. Fix d(n) = ∆, and for the sake of simplicity, let 2 k = 2. The probability that there is a clique of size K = c2 ∆ is larger than n−K , where c2 is the constant in the upper bound of the Main Theorem. Consequently, with at least this probability the chromatic number is at least K > c2 ∆ > c2 d(n)/ log d(n). On the other 2 hand, n−K is not super-polynomially small. Brooks type bound The general Brooks conjecture says that if a graph has maximal degree ∆ and does not have a small clique, then χ(G) is considerably smaller than ∆. This conjecture is a major
14
VAN H. VU
conjecture in the theory of graph coloring and has been considered by various researchers (see [JT] for a survey). Recently, J.H. Kim [Kim] proved that if a graph has girth at least 5 and its maximal degree is ∆, then the (chromatic) choice number is at most c∆/ log ∆. With a very elegant and powerful proof, Johanson [Joh] strengthens this result by showing that the same bound holds for triangle free graphs. Theorem 5.1. (Johansson) If a graph G has maximal degree ∆ and contains no triangle, then there is a positive constant c such that: χl (G) < c∆/ log ∆. This bound is best possible up to a constant factor. Proven by applying Johansson’s method [Vu] , Theorem 2.3 further strengthens Theorem 5.1 by allowing the graph to have many triangles. More important, it suggests that the “triangle free” condition is not something “magical”, but rather a special case of a more general and natural phenomenon. A result similar to Theorem 2.3 can be shown for hypergraphs. A triangle in a (not necessarily uniform) hypergraph H is a triple of vertices x, y, z such that all three pairs (x, y), (y, z) and (z, x) are adjacent. Theorem 5.2. Suppose that the hypergraph H has maximal degree ∆ and each pair of adjacent vertices is contained in at most ∆1−α triangles, for some constant α. Then there is a constant c such that χl (H) ≤ c∆/ log ∆. Weak choice numbers In this paper we consider the strong proper coloring, which requires that the vertices in an edge must have different colors. Another popular definition of a proper coloring is the so-called weak coloring. In a proper weak coloring, every edge must have at least two colors, that is, no edge is monochromatic. For graphs, the notions of strong and weak coloring are the same; but for hypergraphs, they are quite different. Using the notion of weak proper coloring, we can define weak chromatic number and weak choice number accordingly. The asymptotic behavior of the weak chromatic number of random hypergraphs is determined in [KS]. In a forthcomming paper [KrV], Krivelevich and the present author compute the sharp order of magnitude of the weak choice number of random hypergraphs. It turns out that if a random hypergraph is sufficiently dense, then its weak chromatic number and weak choice number are asymptotically the same. Open questions. • It is proved that there are positive constants c1 and c2 such that c1 < χl (H(k, n, p))/χ(H(k, n, p)) < c2 almost surely. It remains an open problem to determine the best value of c1 and c2 . It is clear that one can set c1 = 1, and it seems plausible that c2 = 1 + o(1). However, our approach, in particular the methods used to prove Theorem 2.1-2.3 are not strong enough to prove this. Very recently, Krivelevich and Sudakov computed the sharp values of the constants c1 and c2 in Theorem 1.3 [KS]. It turned out that one can set c1 = 1/2 − o(1) and c2 = 1/2 + o(1), for all value of p such that d(n) = O(n). We conjecture that the
ON THE CHOICE NUMBER OF RANDOM HYPERGRAPHS
15
same constants can be used for Main Theorem. It seems that the method used in [KS] could be modified to prove the conjecture for the case the random hypergraph is dense (d(n) > nf (k) , for some f (k)). However when d(n) is small the conjecture appears to be hard. • The problem of determining the concentration of χ(G(n, p)) and χl (G(n, p)) obtains considerable attention. It was shown by L Ã uczak [ÃLuc2] that if p < n−5/6 then χ(G(n, p)) concentrates on only two values. This result was strengthened by Alon and Krivelevich, who proved that the same result holds for p < n−1/2−² , for any positive constant ². Using the approach in [ÃLuc2] and [AK], it was shown in [AKS] that if p < n−3/4−² then χl (G(n, p)) also concentrates on two values. We conjecture that if d(n) is a constant, then the situation is even better, i.e., χl (G(n, p)) (and more generally χl (H(k, n, p)) ) concentrates on only one value. We say that d is ²-good if there is a number r such that P r(χl H(k, n, p) = r) > 1 − ² if n is sufficiently large, where p = p(n) is chosen so that d(n) = pnk−1 /(k − 2)! = d. Conjecture 5.3. For any ² > 0, the set D = {d|d not ² -good} has zero Lebesgue measure. This conjecture is supported by the following result, proved by Achlioptas and Friedgut [AF]. Theorem 5.4. Let ² > 0 be some constant. For a given n, define a number x > 1 to be bad if the chromatic number of G(n, x/n) is not determined with probability greater than 1 − ², that is, ∀k, Pr[χ(G(n, x/n)) = k] ≤ 1 − ². For a given bound y > 1 define X²y = {x : y > x > 1, and x is bad} . Then for every ² > 0, and every y, lim m(X²y ) = 0
n→∞
where m is the Lebesgue measure. Acknowledgement. We would like to thank the referee for his careful reading and useful comments. REFERENCES [AF] D. Achlioptas and E. Friedgut, A sharp threshold for k-colorability, preprint. [Alo] N. Alon, Restricted colorings of graphs, Surveys in Combinatorics, Proc 14th British Combinatorial Conference, London Mathematical Society Lecture Notes Series 187, edited by K. Walker, Cambridge Univ. Press, 1993, 1-33. [AK] N. Alon and M. Krivelevich, On the concentration of the chromatic number of random graphs, Combinatorica 17 (1997), 401-426. [AKS] N. Alon, M. Krivelevich and Sudakov, List coloring of random and pseudo-random graphs, submitted. [AS] N. Alon and J. Spencer, The Probabilistic Method, Wiley, New York, 1992. [Bol] B. Bollob´as, Random Graphs, Academic Press, London, 1985. [Bol2] B. Bollob´as, The chromatic number of random graphs, Combinatorica 8 (1988), 49–55.
16
VAN H. VU
[ERT] P. Erd˝os, A. L. Rubin and H. Taylor, Choosability in graphs, Proc. West Coast Conf. on Combinatorics, Graph Theory and Computing, Congressus Numerantium XXVI, 1979, 125–157. [FM] A. Frieze and C.J.H. McDiarmid, Algorithmic theory of random graphs, Random structures and algorithms 10 (1997), 5–42. [GM] G. R. Grimmett and C.J.H. McDiarmid, On coloring random graphs, Proccedings of the Cambridge Philosophial Society 77 (1971) 313–324. [Joh] A. Johansson, Asymptotic choice number for triangle free graphs, DIMACS Technical Report 91–95 (1996). [JT] T.R. Jensen and B. Toft, Graph coloring problems , Wiley, 1995. [KL] M. Karonski and T. L Ã uczak, Random hypergraphs, Erd˝ os is eighty (Vol 2), 283-293. [Kim] J. H. Kim, On Brooks’ theorem for sparse graphs, Combinatorics, Probability and Computing 4 (1995), 97–132. [KV] J.H. Kim and V.H. Vu, Small complete arcs of finite projective planes, preprint. [KM] A. Kostochka and N. Mazurova, An inequality in the theory of graph coloring, Diskret. Analiz. 30 (1977), 23–29. [KrV] M. Krivelevich and V.H. Vu, The weak choice number of random hypergraphs, in preparation. [KS] M. Krivelevich and B. Sudakov, The chromatic number of random hypergraphs, Random structures and algorithms 12 (1998), 381-402. [Lov] L. Lov´ asz, Problem and Exercises in Combinatorics, North Holland 1979, chapter 7. [ÃLuc], T. L Ã uczak, The chromatic number of random graphs, Combinatorica 11 (1991), 45-54. [ÃLuc2] T. L Ã uczak, A note on the sharp concentration of the chromatic number of random graphs, Combinatorica 11 (1991), 295–297. [Mat] D.W. Matula, Expose-and-Merge exploration and the chromatic number of random graphs, Combinatorica 9 (1987), 275–284. [Sch] J. Schmidt, Probabilistic analysis of strong hypergraph coloring algorithm and the strong chromatic number, Discrete Math. 66 (1987), 259-277. [Sha] E. Shamir, Chromatic numbers of random hypergraphs and associated graphs, Advances in computing research 5 (1989) 127-142. [SSU] J. Schmidt-Pruzan, E. Shamir and E. Upfal, Random hypergraph coloring and the weak chromatic number, Journal of Graph Theory 9 (1985), 347-362. [Viz] V. G. Vizing, Coloring the vertices of a graph in prescribed colors (in Russian), Diskret. Analiz. No. 29, Metody Diskret. Anal. v. Teorii Kodov i Shem 101 (1976), 3–10. [Vu] Van H. Vu, On some simple degree conditions which guarantee the upper bound on the chromatic (choice) number of random graphs, submitted to Journal of Graph Theory.
