Bounding χ by a fraction of Δ for graphs without large cliques

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Abstract

The greedy coloring algorithm shows that a graph of maximum degree at most Δ has chromatic number at most Δ+1, and this is tight for cliques. Much attention has been devoted to improving this “greedy bound” for graphs without large cliques. Brooks famously proved that this bound can be improved by one if Δ3 and the graph contains no clique of size Δ+1. Reed's Conjecture states that the “greedy bound” can be improved by k if the graph contains no clique of size Δ+12k. Johansson proved that the “greedy bound” can be improved by a factor of Ω(ln(Δ)1) or Ω(ln(ln(Δ))ln(Δ)) for graphs with no triangles or no cliques of any fixed size, respectively.

Notably missing is a linear improvement on the “greedy bound” for graphs without large cliques. In this paper, we prove that for sufficiently large Δ, if G is a graph with maximum degree at most Δ and no clique of size ω, thenχ(G)72Δln(ω)ln(Δ). This implies that for sufficiently large Δ, if ω(72c)2Δ then χ(G)Δ/c.

This bound actually holds for the list-chromatic and even the correspondence chromatic number (also known as DP-chromatic number). In fact, we prove what we call a “local version” of it, a result implying the existence of a coloring when the number of available colors for each vertex depends on local parameters, like the degree and the clique number of its neighborhood. We prove that for sufficiently large Δ, if G is a graph of maximum degree at most Δ and minimum degree at least ln2(Δ) with list-assignment L, then G is L-colorable if for each vV(G),|L(v)|72deg(v)min{ln(ω(v))ln(deg(v)),ω(v)ln(ln(deg(v)))ln(deg(v)),log2(χ(v)+1)ln(deg(v))}, where χ(v) denotes the chromatic number of the neighborhood of v and ω(v) denotes the size of a largest clique containing v. This simultaneously implies the linear improvement over the “greedy bound” and the two aforementioned results of Johansson.

Introduction

Let G be a graph, and for each vV(G), let L(v)N be a set which we call the available colors for v. If each set L(v) is non-empty, then we say that L is a list-assignment for G. If k is a positive integer and |L(v)|k for every vV(G), then we say that L is a k-list-assignment for G. An L-coloring of G is a mapping ϕ with domain V(G) such that ϕ(v)L(v) for every vV(G) and ϕ(u)ϕ(v) for every pair of adjacent vertices u,vV(G). 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 L(v)={1,,k} for every vV(G), 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 χ(G), is the smallest k such that G is k-colorable. The list-chromatic number of G, denoted χ(G), 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 χ(G)=O(Δln(Δ)). 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 Δ2ln(Δ). In 1995, Kim [17] proved that the upper bound holds with a leading constant of 1+o(1) 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 4+o(1), and in 2019, Molloy [19], in the following theorem, improved it to 1+o(1), matching the bound of Kim.

Theorem 1.1 Molloy [19]

If G is a triangle-free graph of maximum degree at most Δ, thenχ(G)(1+o(1))Δln(Δ).

Johansson [16] also proved that for any fixed ω4, if G is a graph of maximum degree at most Δ with no clique of size greater than ω, then χ(G)=O(Δln(ln(Δ))ln(Δ)); 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, thenχ(G)200ωΔln(ln(Δ))ln(Δ).

In 1998, Reed [22] conjectured the following, sometimes referred to as “Reed's ω,Δ,χ 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, thenχ(G)12(Δ+1+ω).

It is possible that Conjecture 1.3 is also true for the list-chromatic number. As evidence for his conjecture, Reed [22] proved the following.

Theorem 1.4 Reed [22]

There exists ε>0 such that the following holds. If G is a graph of maximum degree at most Δ with no clique of size greater than ω, thenχ(G)(1ε)(Δ+1)+εω.

Note that Theorem 1.4 holds for ε=12 if and only if Conjecture 1.3 is true. In 2016, Bonamy, Perrett, and Postle [7] proved that Theorem 1.4 holds for ε=126 when Δ is sufficiently large, and Delcourt and Postle [11] proved that Theorem 1.4 holds for the list-chromatic number for ε=113 in this case. Recently, Hurley, de Joannis de Verclos, and Kang [14] proved that Theorem 1.4 holds for ε=19 when Δ is sufficiently large. Results from Ramsey Theory imply that Theorem 1.4 is not true for any value of ε>12; for example, Spencer [24] showed the existence of a graph on n vertices with independence number 2 (and thus chromatic number at least n/2) such that every clique has size at most n12+o(1). 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 ε>12, and even that the rounding in Conjecture 1.3 is necessary.

Theorem 1.2 implies Conjecture 1.3 when ω=o(ln(Δ)ln(ln(Δ))). It is natural to ask if a bound stronger than that of Conjecture 1.3 can be proved if ω=o(Δ) even if ωln(Δ)200ln(ln(Δ)). Spencer's result implies that the bound cannot be improved if ω=Ω(Δ1/2). Considering this, we were motivated to answer the following question.

Question 1.5

Does there exist a function f:RR+ such that, for every c>1 and every graph G of maximum degree at most Δ with no clique of size greater than Δ1/f(c), we have χ(G)Δ/c?

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, thenχ(G)=O(Δln(ω)ln(Δ)).

Theorem 1.6 answers Question 1.5 in the affirmative with a function f(c) that is quadratic in c; for large enough Δ, the function f(c)=(72c)2 suffices. Determining the best possible function f that confirms Question 1.5 would be very interesting. As mentioned, Spencer [24] showed that f(2)2. This result actually provides a lower bound on f(c) that is linear in c. Spencer [24] proved that the Ramsey number R(c,ω) is at least Ω((ω/ln(ω))c+12) as ω for fixed c3. Therefore there exists a graph G on n vertices with no independent set of size c (and thus chromatic number at least n/(c1)) and no clique of size ω where n is at least ωc+12o(1). Since the maximum degree of a graph is at most its number of vertices, it follows that f(c)c/2+1 if cN.

The bound of Spencer [24] was improved by Kim in [18] for c=3 by a factor of lnω (matching the upper bound of Ajtai, Komlós, and Szemerédi [1] up to a constant factor), by Bohman in [5] for c=4 by a factor of lnω, and by Bohman and Keevash in [6] for c5 by a factor of ln1c2ω, but these improvements do not change the resulting lower bound on f(c).

Since every graph G satisfies α(G)χ(G)|V(G)|, where α(G) 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, thenα(G)Ω(nΔlnωlnΔ).

We actually prove a result stronger than Theorem 1.6. One might wonder if the bounds on |L(v)| 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 deg(v) denote the degree of v, ω(v) denote the size of a largest clique containing v, and χ(v) 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 |L(v)|f(v) where f(v) depends on parameters such as deg(v) and ω(v). 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 |L(v)|=deg(v) 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 (L,M). For each vV(G), letf(v)=72min{ln(ω(v))ln(deg(v)),ω(v)ln(ln(deg(v)))ln(deg(v)),log2(χ(v)+1)ln(deg(v))}. If for each vV(G),|L(v)|deg(v)f(v) and deg(v)ln2(Δ), then G is (L,M)-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 (L,M) such that for each vV(G),|L(v)|Cdeg(v)ln(deg(v)), and deg(v)ln2(Δ), then G is (L,M)-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 (L,M) such that for each vV(G),|L(v)|Cdeg(v)ω(v)ln(ln(deg(v)))ln(deg(v)), and deg(v)ln2(Δ), then G is (L,M)-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 (L,M) such that for each vV(G),|L(v)|Cdeg(v)log2(r+1)ln(deg(v)), and deg(v)ln2(Δ), then G is (L,M)-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 (L,M) such that for each vV(G),|L(v)|Cdeg(v)ln(ω(v))ln(deg(v)), and deg(v)ln2(Δ), then G is (L,M)-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 ln2(Δ), then Corollary 1.11 implies χ(G)CΔln(ω)ln(Δ), 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 G, having G as a subgraph, and with minimum degree at least ln2(Δ). The result now follows by applying Corollary 1.11 to G. 

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 4ln(2), as follows.

Theorem 1.12

For every ξ>0, if Δ is sufficiently large and G is a triangle-free graph of maximum degree at most Δ with correspondence assignment (L,M) such that for each vV(G),|L(v)|(4+ξ)deg(v)log2(deg(v)) and deg(v)ln2(Δ), then G is (L,M)-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 α(H) and i(H) 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 (L,M), and ε(0,1/2). Let ,t:V(G)N, and for each vV(G), let αmin(v) be the minimum of α(H) taken over all subgraphs HG[N(v)] such that i(H)t(v). If for each vV(G),|L(v)|max{2deg(v)(1ε)2αmin(v),2t(v)(v)ε}, and

  • 1.

    ε(1ε)(v)t(v)18lnΔ+6ln16,

  • 2.

    (v)36lnΔ+12ln16, and

  • 3.

    (deg(v)(v))/(v)!<Δ3/8,

then G is (L,M)-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 (deg(v)(v))/(v)!<Δ3/8 limits our choice of (v) to satisfy (v)deg(v). In our applications, our bound on αmin(v) is increasing as a function of t(v), so it is best to choose t(v) as large as possible. Since we require |L(v)|2t(v)(v)/ε, we need t(v)deg(v), but it may be useful to think of t(v) and (v) as being close to deg(v). The function f(v) in Theorem 1.7 corresponds to 2(1ε)2/αmin(v).

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 αmin(v). We do this by proving a general bound on α(H) for a graph H in terms of i(H) and ω(H). For large values of ω(H), 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 (L,M)-coloring of G uniformly at random, and we show that with high probability each uncolored vertex v has at least (v) remaining colors and at most (v) uncolored neighbors u such that (u)(v). 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 E(G) where for each e=uvE(G), Me is a matching of {u}×L(u) and {v}×L(v), we say (L,M) is a correspondence-assignment for G.

  • An (L,M)-coloring of G is a function ϕ:V(G)N such that ϕ(u)L(u) for every uV(G), and for every e=uvE(G), (u,ϕ(u))(v,ϕ(v))Me. If G has an (L,M)-coloring, then we say G is (L,M)-colorable.

A correspondence-assignment (L,M) is a

Proof of Theorem 1.13

In this section, we prove Theorem 1.13. We assume G,(L,M),Δ,, 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.

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