A local epsilon version of Reed's Conjecture

11/06/2019 ∙ by Tom Kelly, et al. ∙ 0

In 1998, Reed conjectured that every graph G satisfies χ(G) ≤1/2(Δ(G) + 1 + ω(G)), where χ(G) is the chromatic number of G, Δ(G) is the maximum degree of G, and ω(G) is the clique number of G. As evidence for his conjecture, he proved an "epsilon version" of it, i.e. that there exists some ε > 0 such that χ(G) ≤ (1 - ε)(Δ(G) + 1) + εω(G). It is natural to ask if Reed's conjecture or an epsilon version of it is true for the list-chromatic number. In this paper we consider a "local version" of the list-coloring version of Reed's conjecture. Namely, we conjecture that if G is a graph with list-assignment L such that for each vertex v of G, |L(v)| ≥1/2(d(v) + 1 + ω(v)), where d(v) is the degree of v and ω(v) is the size of the largest clique containing v, then G is L-colorable. Our main result is that an "epsilon version" of this conjecture is true, under some mild assumptions. Using this result, we also prove a significantly improved lower bound on the density of k-critical graphs with clique number less than k/2, as follows. For every α > 0, if ε≤α^2/1350, then if G is an L-critical graph for some k-list-assignment L such that ω(G) < (1/2 - α)k and k is sufficiently large, then G has average degree at least (1 + ε)k. This implies that for every α > 0, there exists ε > 0 such that if G is a graph with ω(G)≤ (1/2 - α)mad(G), where mad(G) is the maximum average degree of G, then χ_ℓ(G) ≤ (1 - ε)(mad(G) + 1) + εω(G).

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1 Introduction

Let be a graph, and let be a collection of lists which we call available colors. If each set is non-empty, then we say that is a list-assignment for . If is an integer and for every , then we say that is a -list-assignment for . An -coloring of is a mapping with domain such that for every and for every pair of adjacent vertices . If has an -coloring, then we say is -colorable. We say that is -list-colorable, or -choosable, if has an -coloring for every -list-assignment . If for every , then we call an -coloring of a -coloring, and we say is -colorable if has a -coloring. The chromatic number of , denoted , is the smallest such that is -colorable. The list-chromatic number of , denoted , is the smallest such that is -list-colorable.

It is easy to see that for every graph ,

(1)

where denotes the size of a largest clique in and denotes the maximum degree of a vertex in . If

is a clique or an odd cycle then the upper bound in (

1) is tight for both the chromatic number and list-chromatic number. A classical theorem of Brooks [2] says that for the chromatic number, this is essentially the only case in which it is tight. [Brooks’ Theorem [2]] If is a connected graph that is not a clique or odd cycle, then .

In 1998, Reed [20] famously conjectured that, up to rounding, the chromatic number of a graph is at most the average of its clique number and maximum degree plus one. [Reed’s Conjecture [20]] For every graph ,

As evidence for his conjecture, Reed [20] proved that the chromatic nuumber of a graph is at most a weighted average of its clique number and maximum degree plus one. We call this an “epsilon version” of Reed’s Conjecture. [Reed [20]] There exists such that for every graph ,

Reed [20] originally proved that Theorem 1 holds for graphs of sufficiently large maximum degree for . In 2016, Bonamy, Perrett, and Postle [1] improved this to . Recently, Delcourt and Postle [5] (see [6] for an extended abstract) improved this further to . The blowup of a 5-cycle demonstrates that Theorem 1 does not hold for and that the rounding in Reed’s Conjecture is necessary.

It is natural to wonder if Brooks’ Theorem or even Reed’s Conjecture is true for the list-chromatic number. We conjecture that for Reed’s Conjecture this is the case. For every graph ,

The result of Delcourt and Postle [5] is actually proved for the list-chromatic number, implying an “epsilon version” of Conjecture 1.

In 1979, in one of the papers that introduced list-coloring, Erdős, Rubin, and Taylor [8] proved the following classical theorem. [Erdős, Rubin, and Taylor [8]] Let be a connected graph with list-assignment . If for every , , then is -colorable, unless every block of is a clique or an odd cycle and for every , .

Note that Theorem 1 implies that Brooks’ Theorem is true for the list-chromatic number. We consider Theorem 1 to be the archetype of what we call a “local version.” The main focus of this paper is the following conjecture, which we consider to be the natural “local version” of Reed’s Conjecture and Conjecture 1.

[Local Version of Reed’s Conjecture] If is a graph with list-assignment such that for every ,

where is the size of the largest clique containing , then is -colorable.

Note that if true, Conjecture 1 implies Reed’s Conjecture and Conjecture 1. As evidence for Conjecture 1, we prove an “epsilon version” of it, under certain mild assumptions. The following is the main result of this paper. Let . If is a graph of sufficiently large maximum degree and is a list-assignment for such that for all , and

then is -colorable.

We prove Theorem 1 by proving structural properties of a “minimum counterexample” that enable us to then find an -coloring using the probabilistic method. The assumption in Theorem 1 that for each vertex , , implies both that no vertex has a neighborhood that is “too close” to being a clique and that the minimum number of available colors for a vertex is sufficiently large. As we will see, this in turn implies that a minimum counterexample to Theorem 1 has sufficiently large minimum degree. It would be interesting to prove Theorem 1 with the hypothesis that for each vertex replaced with the weaker assumption that the minimum degree of is at least . As we discuss in Section 2, this may be possible to prove with an extension of our methods, at the expense of a worse value of . However, since we use the probabilistic method, we do not believe our techniques could be extended to to prove Conjecture 1 in full.

In Section 2, we provide an overview of the proof of Theorem 1, and Sections 3-6 are devoted to its proof. In order to prove Theorem 1, we needed to develop a new version of Talagrand’s “Concentration Inequality,” Theorem 6, which we prove in Appendix A. Our proof of Theorem 6 corrects a flaw in a version of Talagrand’s Inequality in the book of Molloy and Reed [15, Talagrand’s Inequality II] (see Remark 6 in Section 6).

We now discuss some applications of our result.

1.1 King’s Conjecture

In 2009, King [12] conjectured the following strengthening of Reed’s Conjecture. [King [12]] For every graph ,

King’s idea behind Conjecture 1.1 was that a strengthened form of Reed’s Conjecture may be easier to prove using induction. For certain classes of graphs, this idea has been useful. Using this and the structure theory of claw-free graphs of Chudnovsky and Seymour, King [12] proved that Reed’s Conjecture is true for claw-free graphs. The proof also appears in [11]. In 2013, Chudnovsky et al. [4] proved that King’s Conjecture holds for quasi-line graphs, and in 2015 King and Reed [11] proved it for claw-free graphs with a 3-colorable complement.

Note that Conjecture 1, if true, implies Conjecture 1.1, even for list-coloring. The first application of our main result is that it implies that an “epsilon version” of Conjecture 1.1 is true, assuming does not contain a clique of size within a factor of of the maximum degree of . The following corollary follows easily from Theorem 1.

Let . If is a graph of sufficiently large maximum degree such that , then

1.2 Critical Graphs

Now we discuss an application of Theorem 1 to critical graphs. A graph is -critical if is not -colorable but every proper induced subgraph of is, and if is a list-assignment for , then is -critical if is not -colorable but every proper induced subgraph of is. A list-assignment is -uniform if for every vertex , . We denote the average degree of a graph by . The average degree of critical graphs has been extensively studied. Note that a -critical graph has no vertex of degree less than , so the average degree of a -critical graph is trivially at least . Much work has been devoted to improving this bound. In a breakthrough result from 2014, Kostochka and Yancey [14] proved the following lower bound on the number of edges in -critical graphs.

[Kostochka and Yancey [14]] If and is -critical, then

Theorem 1.2 implies the following asymptotic lower bound on the average degree of -critical graphs. [Kostochka and Yancey [14]] Let , and let be a -critical graph on vertices. Then as approaches infinity,

Theorem 1.2 is tight for every for an infinite family of graphs, as shown by Ore [18]. Therefore the asymptotic bound in Corollary 1.2 can not be improved. Kostochka and Yancey asked if their bound can be improved by excluding certain subgraphs, such as cliques, and if similar results hold for list-coloring. This was considered earlier by Kostochka and Stiebitz [13].

[Kostochka and Stiebitz [13]] For every fixed , if is -critical for some -uniform list-assignment and , then

It is natural to not only consider graphs with bounded clique number but also graphs with clique number bounded by a function of . Theorem 1 implies that the bound in Corollary 1.2 can be improved for large if is an -critical graph for some -list-assignment and has no clique of size at least , as follows.

For every , if then the following holds. If is an -critical graph for some -list-assignment such that and is sufficiently large, then

1.3 Maximum Average Degree

The bound on the chromatic number supplied by Reed’s Conjecture can be viewed as the average of the lower and upper bounds provided in (1), as previously mentioned. However, the upper bound in (1) can easily be improved by replacing with , where , the maximum average degree of . In the spirit of Reed’s Conjecture, we conjecture the following which, if true, implies Reed’s Conjecture.

For every graph ,

Note that Conjecture 1.3, if true, would be tight for , since . More generally, and , so the graphs provide an infinite family for which the difference of the right and left side of the inequality in Conjecture 1.3 is at most one.

Another application of Theorem 1 is an “epsilon version” of Conjecture 1.3 for graphs with clique number less than half their maximum average degree. For every , there exists such that the following holds. For every graph such that ,

Theorem 1.3 follows easily from Theorem 1.2. We include the proof in Section 7.

1.4 -minor free graphs

We conclude this section with an application of Theorem 1.2 to Hadwiger’s conjecture, which is considered one of the most important open problems in graph theory. Hadwiger [9] conjectured in 1943 that if a graph has no -minor, then it has chromatic number at most . The best known upper bound on the chromatic number of -minor free graphs to date uses the fact that the chromatic number of a graph is at most its maximum average degree, combined with the following theorem of Thomason [23] providing a tight upper bound on the average degree of -minor free graphs.111Following acceptance of this paper for publication in JCTB, further improvements were made by Norin and Song [17] and Postle [19]. [Thomason [23]] If is a graph with no -minor, then

where is an explicit constant.

By combining Theorem 1.2 with Theorem 1, we can improve the best known upper bound on the chromatic number of -minor free graphs by a constant factor, as follows. If is a graph with no -minor, then

where is the explicit constant from Theorem 1.

Proof.

It suffices to show that for every , if then for sufficiently large , every -minor free graph is -list-colorable. Suppose not. Then there exists a graph with no -minor that is -critical for some -list-assignment where . Using Theorem 1, we may assume .

Let and . Since , . Since is -critical, by Theorem 1.2, . But , a contradiction. ∎

2 Overview of the Proof of Theorem 1 and Outline of the Paper

The following definition will be useful. Let be a graph. For each we let

and if is a list-assignment for , we let

If the graph or list-assignment is clear from the context, we may omit the subscript or in and , respectively. Note that the conditions of Theorem 1 imply that for each vertex , and .

First we discuss our strategy for proving Theorem 1. We use a variant of a technique called the “naive coloring procedure,” given its name by Molloy and Reed [15]

. Essentially, we analyze a random partial coloring of a graph and prove that with nonzero probability this partial coloring can be extended deterministically to a coloring of the whole graph. The random partial coloring is described formally in Definition 

3. After the random partial coloring, we let be the subgraph induced by on the vertices that are not colored, and we let be a list-assignment for so that any -coloring of can be combined with the random partial coloring to obtain an -coloring of . We prove that with nonzero probability is -colorable. To do this, we would like to show that with high probability, for every vertex , , i.e. that . However, this is not the case. In fact, it may be likely that . For example, the neighborhood of a vertex may form cliques, while for the list-assignment , the vertices in each clique have the same list of available colors and vertices in different cliques have disjoint lists of available colors. Nevertheless, if a vertex has many neighbors with at least as many available colors, we are able to show that . This motivates the following definitions.

Let be some constant to be determined later. Let be a graph with list-assignment , let , and let .

  • If , then we say is a subservient neighbor of .

  • If , then we say is an egalitarian neighbor of .

  • If , then we say is a lordlier neighbor of .

For convenience, we will let denote the set of lordlier neighbors of , denote the set of egalitarian neighbors of , and denote the set of subservient neighbors of .

Let be some constant to be determined later. Let be a graph with list-assignment , let , and let be an egalitarian neighbor of .

  • If , then we say is a strongly egalitarian neighbor of .

  • If , then we say is a weakly egalitarian neighbor of .

For convenience, we will let denote the set of strongly egalitarian neighbors of , denote the set of weakly egalitarian neighbors of , and .

If a vertex has many subservient neighbors, then we say is lordly. The names “subservient”, “egalitarian”, and “lordlier” neighbors are evocative of feudalism in medeival Europe, where power is analogous to list size. As mentioned previously, if is a lordly vertex, we are unable to guarantee that for certain list-assignments for ’s subservient neighbors. We resolve this issue by coloring vertices before their subservient neighbors when finding an -coloring, thus giving “priority” to the lordly vertices.

A lordlier neighbor also has the power to choose from more colors. If has many lordlier neighbors or weakly egalitarian neighbors, then it is likely that after the random partial coloring has many neighbors receiving a color not in . If has many egalitarian neighbors, then it is likely that after the random partial coloring there are many colors assigned to multiple neighbors of . In both cases, .

A common technique in coloring is to attempt to greedily color a vertex of smallest degree, since fewer neighbors means fewer potential color conflicts. However, for our “local version,” this technique is not so useful because vertices of lower degree also have fewer available colors. Our trick to finding an -coloring of is to order the vertices of by the size of their list in , from greatest to least, and color greedily, which may seem counterintuitive. This works because we are able to guarantee for every vertex , that is smaller than the number of neighbors of in that will be colored after in this ordering, and thus is larger than the number of neighbors of in that will be colored before in this ordering.

For each vertex , after an application of our naive coloring procedure, we refer to the number of neighbors of receiving a color not in , plus the multiplicity less 1 of each color in assigned to multiple neighbors, plus the number of uncolored subservient neighbors of as the “savings” for . In order to prove Theorem 1, we first prove Theorem 3, which essentially says that it suffices to show that the expected savings for each vertex is at least and is also sufficiently large. Here “sufficiently large” means , which we need in order to show that the savings for each vertex is sufficiently close to its expectation with probability inverse to a polynomial in , in which case we can apply the Lovász Local Lemma to guarantee an outcome for which the savings for every vertex is close to its expectation.

Using Theorem 3 it suffices to show that the expected savings for a vertex is . If the savings for each vertex is at least , then the condition in Theorem 1 that guarantees that the savings for is at least if is small enough. Moreover, the technical condition in Theorem 1 that ensures that the savings are large enough to obtain concentration. In order to prove Theorem 1 without this latter condition with our methods, it is necessary to find a way to show that a vertex with still has savings at least on the order of .

2.1 Outline of the Paper

We prove Theorem 1.2 and Theorem 1.3 in Section 7. The rest of the paper is devoted to the proof of Theorem 1.

In Section 3, we formalize the previous discussion on the “naive coloring procedure” and prove Theorem 3, which could be considered a “metatheorem.” We also use Theorem 3 in a follow-up paper [10]. In order to prove Theorem 3, we need to show that the savings for each vertex is concentrated around its expectation. Lemma 3 makes this precise. We prove Lemma 3 in Section 6.

Before proving Theorem 1, in Section 4 we prove that a minimum counterexample to Theorem 1 has some desirable structure. The main result of Section 4 is Theorem 4, which says that in a minimum counterexample , each either has many non-adjacent egalitarian neighbors, many lordlier neighbors, or many subservient neighbors. The idea to separate the strongly egalitarian neighbors from the weakly egalitiarian neighbors is crucial here, because the weakly egalitarian neighbors of a vertex are also likely to receive a color not in .

In Section 5, we exploit this structure to lower bound the expected value of each type of savings in Lemmas 5, 5, and 5. Using these lemmas in conjunction with Theorem 3, we prove Theorem 1 in Section 5.

In Section 6 we prove Lemma 3, which completes the proof of Theorem 3. In order to prove this lemma we needed to develop a new “concentration inequality,” Theorem 6

, which provides sufficient conditions for a random variable to be concentrated around its expectation with high probability. Theorem 

6 is similar to results provided in [3, 15], but those did not work for our purposes. We prove Theorem 6 in Appendix A using Talagrand’s inequality.

As mentioned above, in Section 7 we prove Theorem 1.2 and Theorem 1.3.

3 The Local Naive Coloring Procedure

The main result of this section is Theorem 3, which gives sufficient conditions for our naive coloring procedure to be extended to a coloring of the whole graph. Namely, we need that the expected “savings” for each vertex is at least and is sufficiently large. Before we can state Theorem 3, we need to formalize our naive coloring procedure.

In this section, we let be a graph with list-assignment , , and be a partial ordering of . When we apply Theorem 3 to prove Theorem 1 in Section 5, we let be 0 and for , we have if . We include these parameters because we plan to use Theorem 3 in a follow-up paper in which and is different. In order to demonstrate how will be used, we need the following definition.

For each and , we say is a -egalitarian neighbor of if has at least available colors. We let denote the set of -egalitarian neighbors of .

As we will see in Section 6 and as alluded to in Section 2, we can not prove that the number of colors assigned to multiple neighbors of that are not -egalitarian is concentrated around its expectation.

To simplify our probabilistic analysis, we use a generalization of list-coloring known as correspondence coloring, first introduced by Dvořák and Postle [7]. Using correspondence coloring also helps improve the value of in Theorem 1, because we can assume egalitarian neighbors of a vertex have at least colors in common, thus making it more likely that a color is assigned to more than one of them. Recall that is a list-assignment for .

  • If is a function defined on where for each , is a matching of and , then is a correspondence assignment for . If for each the matching saturates at least one of or , then we say is total.

  • An -coloring of is a function such that for every , and for every , . If has an -coloring, then is -colorable.

One defines a -correspondence assignment and the correspondence chromatic number in the natural way, but we do not need these terms. For convenience, if , , , and , we will just say . Note that if for each and , , then an -coloring is an -coloring.

For the remainder of this section, let be a correspondence assignment for . We will actually define our naive coloring procedure for correspondence coloring. First, we need some definitions.

  • We say a naive partial -coloring of is a pair where such that for every and is a set of uncolored vertices such that is an -coloring of .

  • If is a naive partial -coloring of , for each , let

    and for each , let be the matching induced by on and .

If is a naive partial -coloring of , then we call a vertex uncolored if it is in , and otherwise we call it colored.

The following proposition is self-evident. If is a naive partial -coloring of and is -colorable, then is -colorable.

The following is a variant of the naive coloring procedure, but it is not the one we use in Theorem 3. The local naive random coloring procedure samples a random naive partial -coloring in the following way. For each ,

  1. choose uniformly at random, and

  2. let if there exists such that and .

We also consider the following proposition to be self-evident. If is a random naive partial -coloring sampled using the local naive random coloring procedure, then

Recall that . Let . We need the following proposition. There exists such that the following holds. Let be a random naive partial -coloring sampled using the local naive random coloring procedure. If for each , and has minimum degree at least , then for each ,

Proof.

By Proposition 3,

We let , and the result follows. ∎

The local naive random coloring procedure may be useful for some applications, but it is not sufficient for our purposes. Using Proposition 3, we want to find a naive partial -coloring of such that is -colorable. To do this, it suffices to show that for every vertex , is less than the number of uncolored neighbors of such that . However, it is possible that for each , and . For this reason, we modify the local naive random coloring procedure so that , as follows. If for each , and has minimum degree at least (as in Proposition 3), then the local naive random coloring procedure with -equalizing coin-flips samples a random naive partial -coloring in the following way.

  1. Let be sampled using the local naive random coloring procedure,

  2. for each , let with probability , and

  3. let .

For the remainder of this section, we assume and satisfy the assumptions of Definition 3, and we let be a random naive partial -coloring sampled using the the local naive random coloring procedure with -equalizing coin-flips. The following proposition shows why the -equalizing coin-flips are useful.

For each ,

Proof.

Let as in Definition 3. Note that . By the choice of , we have , and the result follows. ∎

Recall that a vertex is a -egalitarian neighbor of a vertex if . Recall also that is a partial ordering of . We can now formalize what we mean by the “savings” for each vertex, as follows. For each , we define the following random variables.

  • Let count the number of colored neighbors of such that is not matched by .

  • Let and count the number of nonadjacent pairs and triples respectively of colored -egalitarian neighbors of that receive colors that are matched to the same color in .

  • Let count the number of uncolored neighbors of such that .

  • Let .

More precisely, letting denote the set of triangles of a graph , we have that

We are now prepared to state Theorem 3.

For every and , there exists such that the following holds. If is a graph with correspondence-assignment and a partial ordering of such that

  1. ,

  2. has maximum degree at most and minimum degree at least (as in Proposition 3),

  3. and for each ,

  4. , and

  5. ,

then is -colorable

In order to prove Theorem 3, we need the following lemma. Under the conditions of Theorem 3, if is a random naive partial coloring sampled using the local naive random coloring procedure with -equalizing coin-flips, then with nonzero probability every satisfies

(2)

Observe that by the inclusion-exclusion principle, if we let the repetitiveness of color be one less than the number of colored neighbors such that , then undercounts the total repetitiveness of colors assigned to neighbors of . Therefore

(3)

We need to show that with high probability, these random variables are close to their expectation. We make this precise in the following definition.

We say a random variable is -concentrated if

We will use the following lemma to prove Lemma 3. If is sufficiently large, has maximum degree at most , and , then for each , and are -concentrated. We defer the proof of Lemma 3 to Section 6. Lemma 3 is the reason why we need to include the parameter .

To prove Lemma 3, we will also use the Lovász Local Lemma. [Lovász Local Lemma] Let and a finite set of events such that for every ,

  1. , and

  2. is mutually independent of a set of all but at most other events in .

If , then the probability that none of the events in occur is strictly positive.

Now we are ready to prove Lemma 3.

Proof of Lemma 3.

For each , let be the event that (2) does not hold, and let . Note that for each , depends only on trials at vertices at distance at most two from , so if has distance at least five to , then and do not depend on any of the same trials. Therefore each is mutually independent of a set of all but at most events in .

By Lemma 3, it suffices to show that for each , . Let

and let be the event that

By Lemma 3 and the Union Bound, . We claim that , which completes the proof. By (3), it suffices to show

(4)

By the assumption that , Since is sufficiently large, we may assume that Since , (4) holds, which completes the proof. ∎

We conclude this section with the proof of Theorem 3.

Proof of Theorem 3.

By Proposition 3, it suffices to show that is -colorable with nonzero probability. Thus it suffices to show that for some instance of , for each ,

(5)

because then we can color greedily in the ordering provided by , breaking ties arbitrarily. By Lemma 3, we may consider the instance in which each satisfies (2).

Note that for each ,

Therefore if (2) holds, then (5) holds. Thus, is -colorable, as desired. ∎

4 Structure

The main result of this section is Theorem 4, which lower bounds the number of non-adjacent egalitarian neighbors of a vertex in terms of the number of its neighbors that are lordlier, subservient, or weakly egalitarian.

First we need to prove Theorem 4, which may be of independent interest. It bounds the number of edges of a critical graph in terms of the size of a matching in the complement. Recall that a graph with list-assignment is -critical if is not -colorable but every proper induced subgraph of is.

If is -critical, is an induced subgraph of , and is a matching in , then

We will apply Theorem 4 to an appropriate subset of the neighborhood of each vertex. In order to prove Theorem 4, we need an improved version of a classic result of Erdős, Rubin, and Taylor [8] about list-coloring a complete graph with a matching removed, proved by Delcourt and Postle [5]. We include a proof for completeness.

[Delcourt and Postle [5]] If , where is a matching and is a list-assignment for such that

  1. for all , and ,

  2. for all , ,

then is -colorable.

Proof.

We proceed by induction on . If , then and by 2, for all . So we may assume .

Suppose there exists such that . Let , and for all , let . Let and . Then and satisfy conditions 1 and 2. By induction, has an -coloring. Therefore has an -coloring, obtained from an -coloring by coloring and with color , as desired.

Therefore we may assume that for all , . Since , . We claim for all , . If there exists such that , then , as claimed. Therefore we may assume that . If , then , as claimed. Hence, we may assume that . But then , as claimed.

Therefore for all . By Hall’s Theorem, there is a matching from to , and thus has an -coloring, as desired. ∎

Now we prove Theorem 4.

Proof of Theorem 4.

We proceed by induction on . Since is -critical, has an -coloring . For all , let . Since does not have an -coloring, does not have an -coloring. By Lemma 4, either there exists such that or or there exists such that . Note that for all ,

(6)

If there exists such that , then let and . By (6), . Hence,

By induction, . Therefore,

as desired.

If there exists such that