Bounding χ by a fraction of Δ for graphs without large cliques
Introduction
Let G be a graph, and for each , let be a set which we call the available colors for v. If each set is non-empty, then we say that L is a list-assignment for G. If k is a positive integer and for every , then we say that L is a k-list-assignment for G. An L-coloring of G is a mapping ϕ with domain such that for every and for every pair of adjacent vertices . We say that a graph G is k-list-colorable, or k-choosable, if G has an L-coloring for every k-list-assignment L. If for every , then we call an L-coloring of G a k-coloring, and we say G is k-colorable if G has a k-coloring. The chromatic number of G, denoted , is the smallest k such that G is k-colorable. The list-chromatic number of G, denoted , is the smallest k such that G is k-list-colorable.
In 1996, Johansson [15] famously proved that if G is a triangle-free graph of maximum degree at most Δ, then . Determining the best possible value of the leading constant in this bound is of general interest. (According to Molloy and Reed [20], the leading constant in Johansson's proof was 9.) The best known lower bound, using random Δ-regular graphs, is . In 1995, Kim [17] proved that the upper bound holds with a leading constant of for graphs of girth at least five. In 2015, Pettie and Su [21] improved the leading constant in the upper bound for triangle-free graphs to , and in 2019, Molloy [19], in the following theorem, improved it to , matching the bound of Kim. Theorem 1.1 Molloy [19] If G is a triangle-free graph of maximum degree at most Δ, then
Johansson [16] also proved that for any fixed , if G is a graph of maximum degree at most Δ with no clique of size greater than ω, then ; however, the proof was never published. Molloy [19] proved the following stronger result, which holds even when ω is not fixed.
Theorem 1.2 Molloy [19] If G is a graph of maximum degree at most Δ with no clique of size greater than ω, then
In 1998, Reed [22] conjectured the following, sometimes referred to as “Reed's Conjecture.”
Conjecture 1.3 Reed [22] If G is a graph of maximum degree at most Δ with no clique of size greater than ω, then
Theorem 1.4 Reed [22] There exists such that the following holds. If G is a graph of maximum degree at most Δ with no clique of size greater than ω, then
Note that Theorem 1.4 holds for if and only if Conjecture 1.3 is true. In 2016, Bonamy, Perrett, and Postle [7] proved that Theorem 1.4 holds for when Δ is sufficiently large, and Delcourt and Postle [11] proved that Theorem 1.4 holds for the list-chromatic number for in this case. Recently, Hurley, de Joannis de Verclos, and Kang [14] proved that Theorem 1.4 holds for when Δ is sufficiently large. Results from Ramsey Theory imply that Theorem 1.4 is not true for any value of ; for example, Spencer [24] showed the existence of a graph on n vertices with independence number 2 (and thus chromatic number at least ) such that every clique has size at most . The blowup of a 5-cycle, i.e. the Cartesian product of a clique and a 5-cycle, also demonstrates that Theorem 1.4 is not true when , and even that the rounding in Conjecture 1.3 is necessary.
Theorem 1.2 implies Conjecture 1.3 when . It is natural to ask if a bound stronger than that of Conjecture 1.3 can be proved if even if . Spencer's result implies that the bound cannot be improved if . Considering this, we were motivated to answer the following question.
Question 1.5 Does there exist a function such that, for every and every graph G of maximum degree at most Δ with no clique of size greater than , we have ?
Our first result in this paper is the following theorem. Theorem 1.6 If G is a graph of maximum degree at most Δ with no clique of size greater than ω, then
Theorem 1.6 answers Question 1.5 in the affirmative with a function that is quadratic in c; for large enough Δ, the function suffices. Determining the best possible function f that confirms Question 1.5 would be very interesting. As mentioned, Spencer [24] showed that . This result actually provides a lower bound on that is linear in c. Spencer [24] proved that the Ramsey number is at least as for fixed . Therefore there exists a graph G on n vertices with no independent set of size c (and thus chromatic number at least ) and no clique of size ω where n is at least . Since the maximum degree of a graph is at most its number of vertices, it follows that if .
The bound of Spencer [24] was improved by Kim in [18] for by a factor of (matching the upper bound of Ajtai, Komlós, and Szemerédi [1] up to a constant factor), by Bohman in [5] for by a factor of , and by Bohman and Keevash in [6] for by a factor of , but these improvements do not change the resulting lower bound on .
Since every graph G satisfies , where is the size of a largest independent set in G, Theorem 1.6 strengthens the following result of Bansal, Gupta, and Guruganesh [3, Theorem 1.2]: If G is a graph on n vertices of maximum degreee at most Δ with no clique of size greater than ω, then
We actually prove a result stronger than Theorem 1.6. One might wonder if the bounds on supplied by Theorem 1.1, Theorem 1.2, Theorem 1.4, Theorem 1.6 can be relaxed to depend on local parameters, such as the degree of the vertex v, or the size of a largest clique containing v, rather than the global parameters Δ and ω. To that end, for a vertex v, we let denote the degree of v, denote the size of a largest clique containing v, and denote the chromatic number of the (open) neighborhood of v.
We are interested in proving that a graph G is L-colorable whenever every vertex v satisfies where depends on parameters such as and . The archetypal example is the classical theorem of Erdős, Rubin, and Taylor [13] that a graph G is degree-choosable (meaning L-colorable for any list-assignment L satisfying for every vertex v) unless every block in G is a clique or an odd cycle. If the function f is related to a bound on χ in terms of a global graph parameter, then we call such a theorem a “local version” of this bound. Thus, Erdős, Rubin, and Taylor's [13] degree-choosability result is a local version of Brooks' Theorem [8]. Our main result implies local versions of Theorem 1.1, Theorem 1.2, Theorem 1.6 simultaneously, although we do not match the leading constant in Theorem 1.1.
In fact, we prove the theorem for correspondence coloring, a generalization of list-coloring introduced by Dvořák and Postle [12] in 2015, and also known as DP-coloring. We provide a definition in Section 2; the theorem as stated below can also be read as if L is a list assignment.
Theorem 1.7 For all sufficiently large Δ the following holds. Let G be a graph of maximum degree at most Δ with correspondence assignment . For each , let If for each , and , then G is -colorable.
Recently, Bernshteyn [4] proved that Theorem 1.1, Theorem 1.2 hold for the correspondence chromatic number, which is always at least as large as the list-chromatic number. Our Theorem 1.7 implies that “local versions” of these theorems are true for correspondence coloring, as follows.
Corollary 1.8 For some constant C the following holds. If G is a triangle-free graph of maximum degree at most Δ with correspondence assignment such that for each , and , then G is -colorable.
Corollary 1.9 For some constant C the following holds. If G is a graph of maximum degree at most Δ with correspondence assignment such that for each , and , then G is -colorable.
We also derive the following “local version” of a result of Johansson [16] on graphs that are locally r-colorable, meaning the neighborhood of every vertex is r-colorable. Corollary 1.10 For some constant C the following holds. If G is a locally r-colorable graph of maximum degree at most Δ with correspondence assignment such that for each , and , then G is -colorable.
Of course, Theorem 1.7 also implies a “local version” of Theorem 1.6, as follows. Corollary 1.11 For some constant C the following holds. If G is a graph of maximum degree at most Δ with correspondence assignment such that for each , and , then G is -colorable.
We now argue that Theorem 1.6 follows from Corollary 1.11. Proof of Theorem 1.6 assuming Corollary 1.11 Let G be a graph of maximum degree at most Δ with no clique of size greater than ω. We may assume that G has minimum degree at least one. If G has minimum degree at least , then Corollary 1.11 implies , as desired. Otherwise, we use the following standard procedure to obtain a graph of larger minimum degree containing G as a subgraph. We duplicate the graph G, and we add an edge between each vertex of minimum degree and its duplicate. Note that the minimum degree is increased by one, and that for every vertex v, the size of a largest clique containing v in the new graph does not increase. We repeat this procedure until we obtain a graph , having G as a subgraph, and with minimum degree at least . The result now follows by applying Corollary 1.11 to . □
Although we cannot match the leading constant in Theorem 1.1 in our “local version,” we can get the leading constant within a factor of , as follows.
Theorem 1.12 For every , if Δ is sufficiently large and G is a triangle-free graph of maximum degree at most Δ with correspondence assignment such that for each , and , then G is -colorable.
Following our initial preprint version of this paper, Davies, de Joannis de Verclos, Kang, and Pirot [9] matched the leading constant in Theorem 1.1 in a “local version” but at the expense of a larger minimum list-size requirement, and Theorem 1.12 was strengthened by Davies, and Kang, and Pirot, and Sereni [10]. In [9, Proposition 11], it is shown that some minimum list size requirement is necessary in any “local version”, but it is not known what is best possible.
As mentioned, Bernshteyn [4] proved that Theorem 1.1, Theorem 1.2 hold for the correspondence chromatic number. Many aspects of Bernshteyn's proofs are similar to those of Molloy's [19]; however, Bernshteyn's proof is much shorter and simpler. Molloy used a proof technique known as “entropy compression,” which proves that a random algorithm terminates. Bernshteyn cleverly realized that the use of entropy compression in Molloy's proof can be replaced with the Lopsided Lovász Local Lemma, resulting in a substantial simplification of the proof.
Both proofs can be applied in the more general setting of graphs in which the median size of an independent set is somewhat large in comparison to the number of independent sets. We make this precise by extracting a more general theorem from their proofs, and we actually prove a “local version” of it, as follows.
For a graph H, let and denote the median size of an independent set and the number of independent sets in H respectively. Theorem 1.13 Let G be a graph of maximum degree at most Δ with correspondence-assignment , and . Let , and for each , let be the minimum of taken over all subgraphs such that . If for each , and , , and ,
then G is -colorable.
We prove Theorem 1.7, Theorem 1.12 using Theorem 1.13. We think that proving Theorem 1.13 separately makes the proof easier to understand, and we think that Theorem 1.13 may have applications not listed in this paper.
The hypothesis limits our choice of to satisfy . In our applications, our bound on is increasing as a function of , so it is best to choose as large as possible. Since we require , we need , but it may be useful to think of and as being close to . The function in Theorem 1.7 corresponds to .
In Section 4, we prove Theorem 1.7, Theorem 1.12 using Theorem 1.13. In order to apply Theorem 1.13, one needs to find a lower bound on . We do this by proving a general bound on for a graph H in terms of and . For large values of , our bound is better than the bound used by Molloy [19], and this yields the improvement in Theorem 1.6.
In Section 3, we prove Theorem 1.13. The proof is similar to Bernshteyn's proof of Theorem 1.2 from [4]; however, we prove the more general theorem, and some changes are necessary in order to prove the “local version” of it. In the proof of Theorem 1.13, we sample a partial -coloring of G uniformly at random, and we show that with high probability each uncolored vertex v has at least remaining colors and at most uncolored neighbors u such that . We use the Lovász Local Lemma to show that with nonzero probability this outcome holds for every vertex, in which case we can complete the partial coloring greedily.
In Section 2, we formally define correspondence coloring. We also discuss some notation about “partial colorings” and some probabilistic tools that are needed in the proof of Theorem 1.13.
Section snippets
Correspondence coloring
In this subsection we define correspondence coloring.
Definition 2.1 Let G be a graph with list-assignment L. If M is a function defined on where for each , is a matching of and , we say is a correspondence-assignment for G. An -coloring of G is a function such that for every , and for every , . If G has an -coloring, then we say G is -colorable.
A correspondence-assignment is a
Proof of Theorem 1.13
In this section, we prove Theorem 1.13. We assume , and t satisfy the conditions of Theorem 1.13 throughout the section.
Proofs of Theorems 1.7 and 1.12
In this section we prove Theorem 1.7, Theorem 1.12. In this section, log means the base 2 logarithm.
Acknowledgements
We would like to thank an anonymous referee for many helpful comments and in particular for pointing out a mistake in the proof of Lemma 3.2 in a previous version of this paper. We would also like to thank Louis Esperet and Ross Kang for sharing reference [3] with us.
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Partially supported by NSERC under Discovery Grant No. 2019-04304, the Ontario Early Researcher Awards program and the Canada Research Chairs program.