Coloring graphs with forbidden bipartite subgraphs

07/12/2021
by   James Anderson, et al.
Georgia Institute of Technology
0

A conjecture of Alon, Krivelevich, and Sudakov states that, for any graph F, there is a constant c_F > 0 such that if G is an F-free graph of maximum degree Δ, then χ(G) ≤ c_F Δ / logΔ. Alon, Krivelevich, and Sudakov verified this conjecture for a class of graphs F that includes all bipartite graphs. Moreover, it follows from recent work by Davies, Kang, Pirot, and Sereni that if G is K_t,t-free, then χ(G) ≤ (t + o(1)) Δ / logΔ as Δ→∞. We improve this bound to (1+o(1)) Δ/logΔ, making the constant factor independent of t. We further extend our result to the DP-coloring setting (also known as correspondence coloring), introduced by Dvořák and Postle.

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1. Introduction

All graphs in this paper are finite, undirected, and simple. The starting point of our investigation is the following celebrated conjecture of Alon, Krivelevich, and Sudakov:

Conjecture 1.1 (Alon–Krivelevich–Sudakov [AKSConjecture, Conjecture 3.1]).

For every graph , there is a constant such that if is an -free graph of maximum degree , then .

Here we say that is -free if has no subgraph (not necessarily induced) isomorphic to . As long as contains a cycle, the bound in Conjecture 1.1 is best possible up to the value of , since there exist -regular graphs of arbitrarily high girth with [BollobasBound]. On the other hand, the best known general upper bound is due to Johansson [Joh_sparse] (see also [Molloy]), which exceeds the conjectured value by a factor.

Nevertheless, there are some graphs for which Conjecture 1.1 has been verified. Among the earliest results along these lines is the theorem of Kim [Kim95] that if has girth at least (that is, is -free), then . (Here and in what follows indicates a function of that approaches as .) Johansson [Joh_triangle] proved Conjecture 1.1 for ; that is, Johansson showed that if is triangle-free, then for some constant . Johansson’s proof gave the value [MolloyReed, 125], which was later improved to by Pettie and Su [PS15] and, finally, to by Molloy [Molloy], matching Kim’s bound for graphs of girth at least .

In the same paper where they stated Conjecture 1.1, Alon, Krivelevich, and Sudakov verified it for the complete tripartite graph [AKSConjecture, Corollary 2.4]. (Note that the case yields Johansson’s theorem.) Their results give the bound for such , which was recently improved to by Davies, Kang, Pirot, and Sereni [DKPS, §5.6]. Numerous other results related to Conjecture 1.1 can be found in the same paper.

Here we are interested in the case when the forbidden graph is bipartite. It follows from the result of Davies, Kang, Pirot, and Sereni mentioned above that if , then Conjecture 1.1 holds with . We improve this bound to (so only the lower order term actually depends on the graph ):

Theorem 1.2 ().

For every bipartite graph and every , there is such that every -free graph of maximum degree satisfies .

In view of the results in [AKSConjecture] and [DKPS], it is natural to ask if a version of Theorem 1.2 also holds for . We give the affirmative answer in a forthcoming paper [nbhd], where we use some of the techniques developed here to prove that every -free graph satisfies . In other words, we eliminate the dependence on in the constant factor, although we are unable to reduce it all the way to .

Returning to the case of bipartite , we establish an extension of Theorem 1.2 in the context of DP-coloring (also known as correspondence coloring), which was introduced a few years ago by Dvořák and Postle [DPCol]. DP-coloring is a generalization of list coloring. Just as in ordinary list coloring, we assume that every vertex of a graph is given a list of colors to choose from. In contrast to list coloring though, the identifications between the colors in the lists are allowed to vary from edge to edge. That is, each edge is assigned a matching (not necessarily perfect and possibly empty) from to . A proper DP-coloring then is a mapping that assigns a color to each vertex so that whenever , we have . Note that list coloring is indeed a special case of DP-coloring which occurs when the colors “correspond to themselves,” i.e., for each and , we have if and only if .

Formally, it is convenient to describe DP-coloring using an auxiliary graph called a DP-cover of . The definition below is a modified version of the one given in [JMTheorem]:

Definition 1.3.

A DP-cover (or a correspondence cover) of a graph is a pair , where is a graph and is a function such that:

  1. [label=(C0)]

  2. The set forms a partion of .

  3. For each , is an independent set in .

  4. For , , the induced subgraph is a matching; this matching is empty whenever

We refer to the vertices of as colors. For , we let denote the underlying vertex of in , i.e., the unique vertex such that . If two colors , are adjacent in , we say that they correspond to each other and write . An -coloring is a mapping such that for all . Similarly, a partial -coloring is a partial map such that whenever is defined. A (partial) -coloring is proper if the image of is an independent set in , i.e., if for all , such that and are both defined. A DP-cover is -fold for some if for all . The DP-chromatic number of , denoted by , is the smallest such that admits a proper -coloring with respect to every -fold DP-cover .

An interesting feature of DP-coloring is that it allows one to put structural constraints not on the base graph, but on the cover graph instead. For instance, Cambie and Kang [CK] made the following conjecture:

Conjecture 1.4 (Cambie–Kang [Ck, Conjecture 4]).

For every , there is such that the following holds. Let be a triangle-free graph and let be a DP-cover of . If has maximum degree and for all , then admits a proper -coloring.

The conclusion of Conjecture 1.4 is known to hold if is taken to be the maximum degree of rather than of [JMTheorem] (notice that , so a bound on is a stronger assumption than a bound on ). Cambie and Kang [CK, Corollary 3] verified Conjecture 1.4 when is not just triangle-free but bipartite. Amini and Reed [AminiReed] and, independently, Alon and Assadi [Pallette, Proposition 3.2] proved a version of Conjecture 1.4 for list coloring, but with replaced by a larger constant ( in [Pallette]). To the best of our knowledge, it is an open problem to reduce the constant factor to even in the list coloring framework.

Notice that in Cambie and Kang’s conjecture, the base graph is assumed to be triangle-free (which, of course, implies that is triangle-free as well). In principle, it is possible that is triangle-free while is not, and it seems that the conclusion of Conjecture 1.4 could hold even then. We suspect that Conjecture 1.1 should also hold in the following stronger form:

Conjecture 1.5 ().

For every graph , there is a constant such that the following holds. Let be a graph and let be a DP-cover of . If is -free and has maximum degree and if for all , then admits a proper -coloring.

After this discussion, we are now ready to state our main result:

Theorem 1.6 ().

There is a constant such that for every , there is such that the following holds. Suppose that , , satisfy

If is a graph and is a DP-cover of such that:

  1. [label=()]

  2. is -free,

  3. , and

  4. for all ,

then has a proper -coloring.

If is an arbitrary bipartite graph with parts of size and , then an -free graph is also -free. Thus, Theorem 1.6 yields the following result for large enough as a function of , , and :

Corollary 1.7 ().

For every bipartite graph and , there is such that the following holds. Let . Suppose is a DP-cover of such that:

  1. [label=()]

  2. is -free,

  3. , and

  4. for all .

Then has a proper -coloring.

Setting in Corollary 1.7 gives an extension of Theorem 1.2 to DP-coloring:

Corollary 1.8 ().

For every bipartite graph and , there is such that every -free graph with maximum degree satisfies .

We close this introduction with a few words about the proof of Theorem 1.6. To find a proper -coloring of we employ a variant of the so-called “Rödl Nibble” method, in which we randomly color a small portion of and then iteratively repeat the same procedure with the vertices that remain uncolored. (See [Nibble] for a recent survey on this method.) Throughout the iterations, both the maximum degree of the cover graph and the minimum list size are decreasing, but we show that the former is decreasing at a faster rate than the latter. Thus, we eventually arrive at a situation where for all , and then it is easy to complete the coloring. The specific procedure in our proof is essentially the same as the one used by Kim [Kim95] (see also [MolloyReed, Chapter 12]) to bound the chromatic number of graphs of girth at least , suitably modified for the DP-coloring framework. We describe it in detail in §3. The main novelty in our analysis is in the proof of Lemma 4.5, which allows us to control the maximum degree of the cover graph after each iteration. This is the only part of the proof that relies on the assumption that is -free. The proof of Lemma 4.5 involves several technical ingredients, which we explain in §5. In §6, we put the iterative process together and verify that the coloring can be completed.

2. Preliminaries

In this section we outline the main probabilistic tools that will be used in our arguments. We start with the symmetric version of the Lovász Local Lemma.

Theorem 2.1 (Lovász Local Lemma; [MolloyReed, §4]).

Let , , …,

be events in a probability space. Suppose there exists

such that for all we have . Further suppose that each is mutually independent from all but at most other events , for some . If , then with positive probability none of the events , …, occur.

Aside from the Local Lemma, we will require several concentration of measure bounds. The first of these is the Chernoff Bound for binomial random variables. We state the two-tailed version below:

Theorem 2.2 (Chernoff; [MolloyReed, §5]).

Let be a binomial random variable on trials with each trial having probability of success. Then for any , we have

We will also take advantage of two versions of Talagrand’s inequality. The first version is the standard one:

Theorem 2.3 (Talagrand’s Inequality; [MolloyReed, §10.1]).

Let be a non-negative random variable, not identically 0, which is a function of independent trials , …, . Suppose that satisfies the following for some , :

  1. [label=(T0)]

  2. Changing the outcome of any one trial can change by at most .

  3. For any , if then there is a set of at most trials that certify is at least .

Then for any , we have

The second version of Talagrand’s inequality we will use was developed by Bruhn and Joos [ExceptionalTal]. We refer to it as Exceptional Talagrand’s Inequality. In this version, we are allowed to discard a small “exceptional” set of outcomes before constructing certificates.

Theorem 2.4 (Exceptional Talagrand’s Inequality [ExceptionalTal, Theorem 12]).

Let be a non-negative random variable, not identically 0, which is a function of independent trials , …, , and let be the set of outcomes for these trials. Let be a measurable subset, which we shall refer to as the exceptional set. Suppose that satisfies the following for some , :

  1. [label=(ET0)]

  2. For all and every outcome , there is a set of at most trials such that whenever differs from on fewer than of the trials in .

  3. , where .

Then for every , we have:

Finally, we shall use the Kővári–Sós–Turán theorem for -free graphs:

Theorem 2.5 (Kővári–Sós–Turán [Kst]; see also [Kst2]).

Let be a bipartite graph with a bipartition , where , , and . Suppose that does not contain a complete bipartite subgraph with vertices in and vertices in . Then .

3. The Wasteful Coloring Procedure

To prove Theorem 1.6, we will start by showing we can produce a partial -coloring of our graph with desirable properties. Before we do so, we introduce some notation used in the next lemma. When is a partial -coloring of , we define . Given parameters , , , , we define the following functions:

The meaning of this notation will become clear when we describe the randomized coloring procedure we use to prove the following lemma.

Lemma 3.1 ().

There are , with the following property. Let , , , , satisfy:

  1. [label=(0)]

  2. ,

  3. ,

  4. ,

  5. ,

Then whenever is a graph and is a DP-cover of such that

  1. [label=(0),resume]

  2. is -free,

  3. ,

  4. for all ,

there exists a partial proper -coloring and an assignment of subsets to each such that, setting

we get that for all , , and :

To prove Lemma 3.1, we will carry out a variant of the the “Wasteful Coloring Procedure,” as described in [MolloyReed, Chapter 12]. As mentioned in the introduction, essentially the same procedure was used by Kim [Kim95] to bound the chromatic number of graphs of girth at least . We describe this procedure in terms of DP-coloring below:

Wasteful Coloring Procedure

Input: A graph with a DP-cover and a parameter .

Output: A proper partial -coloring and subsets for all .

  1. [wide]

  2. Generate a random subset by putting each vertex into independently with probability . The vertices in are said to be activated.

  3. Independently for each , pick a color uniformly at random. We say that is assigned the color and let be the set of the assigned colors.

  4. For each vertex , let

    We say that the colors in are kept by , and that is the set of kept colors. We also say that the colors in the set are removed.

  5. For each , if , then we set . For all other vertices, is undefined, so we set . We call such vertices uncolored.

  6. For each , we let .

In §§4 and 5, we will show that, with positive probability, the output of the Wasteful Coloring Procedure satisfies the conclusion of Lemma 3.1. With this procedure in mind, we can now provide an intuitive understanding for the functions defined in the beginning of this section. Suppose , satisfy and . If we run the Wasteful Coloring Procedure with these and , then is the probability that a color is kept by (i.e., ), while is approximately the probability that a vertex is uncolored (i.e., ). The details of these calculations are given in §4. Note that, assuming the terms and in the definitions of and are small, we can write

In other words, an application of Lemma 3.1 reduces the ratio roughly by a factor of . In §6 we will show that Lemma 3.1 can be applied iteratively to eventually make the ratio less than, say, , after which the coloring can be completed using the following proposition:

Proposition 3.2 ().

Let be a graph with a -cover such that for every , where is the maximum degree of . Then, there exists a proper -coloring of .

This proposition is standard and proved using the Lovász Local Lemma. Its proof in the DP-coloring framework can be found, e.g., in [JMTheorem, Appendix].

4. Proof of Lemma 3.1

In this section, we present the proof of Lemma 3.1, apart from one technical lemma that will be established in §5. We start with the following proposition which allows us to assume that the given DP-cover of is -regular.

Proposition 4.1 ().

Let be a graph and be a DP-cover of such that and is -free for some , , . Then there exist a graph and a DP-cover of such that the following statements hold:

  • is a subgraph of ,

  • is a subgraph of ,

  • for all , ,

  • is -free,

  • is -regular.

Proof.

Set and let be an -regular graph with girth at least . (Such exists by [GirthRegular1, GirthRegular2].) Without loss of generality, we may assume that , where . Take vertex-disjoint copies of , say , …, , and let be a DP-cover of isomorphic to . Define for every . The graphs and are obtained from the disjoint unions of , …, and , …, respectively by performing the following sequence of operations once for each edge , one edge at a time:

  1. Pick arbitrary vertices and .

  2. Add the edge to and the edge to .

  3. If , remove from .

  4. If , remove from .

Since is -regular, throughout this process the sum decreases exactly times, which implies that the resulting graph is -regular. Furthermore, since has girth at least and is -free, is also -free. Hence, if we define so that for all , then is a DP-cover of satisfying all the requirements. ∎

Suppose , , , , , and a graph with a DP-cover satisfy the conditions of Lemma 3.1. By removing some vertices from if necessary, we may assume that for all . Furthermore, by Proposition 4.1, we may assume that is -regular. Since we may delete all the edges of whose corresponding matchings in are empty, we may also assume that . Suppose we have carried out the Wasteful Coloring Procedure with these and . As in the statement of Lemma 3.1, we let

For each and , we define the random variables

Note that if , then ; similarly, if , then . As in Lemma 3.1, we let . For the ease of notation, we will write to mean , to mean , etc. We will show, for large enough, that:

Lemma 4.2 ().

For all , ,

Lemma 4.3 ().

For all , .

Lemma 4.4 ().

For all , ,

Lemma 4.5 ().

For all ,

Together, these lemmas will allow us to complete the proof of Lemma 3.1, as follows.

Proof of Lemma 3.1.

Take so large that Lemmas 4.24.5 hold. Define the following random events for every vertex and every color :

We will use the Lovász Local Lemma, Theorem 2.1. By Lemma 4.2 and Lemma 4.3, we have:

By Lemma 4.4 and Lemma 4.5, we have:

Let . Note that events and are mutually independent from events of the form and where and is at distance at least from . Since we are assuming that , there are at most vertices in of distance at most from . For each such vertex , there are events corresponding to and the colors in , so we can let . Assuming is large enough, we have

so, by Theorem 2.1, with positive probability none of the events , occur, as desired. ∎

The proofs of Lemmas 4.24.4 are fairly straightforward and similar to the corresponding parts of the argument in the girth- case (see [MolloyReed, Chapter 12]). We present them here.

Proof of Lemma 4.2.

Consider any . We have exactly when , i.e., when no neighbour of is assigned to its underlying vertex. The probability of this event is . By the linearity of expectation, it follows that . ∎

Proof of Lemma 4.3.

It is easier to consider the random variable , the number of colors removed from . We will use Theorem 2.3, Talagrand’s Inequality. Order the colors in for each arbitrarily. Let be the random variable that is equal to if and if and is the -th color in . Then , is a list of independent trials whose outcomes determine . Changing the outcome of any one of these trials can affect at most by . Furthermore, if for some , then this fact can be certified by the outcomes of of these trials. Namely, for each removed color , we take the trial corresponding to any vertex such that and is adjacent to in . Thus, we can now apply Theorem 2.3 with , to get:

where the first and last inequalities hold for large enough. ∎

Proof of Lemma 4.4.

Let and . We need to bound the probability that and . We split into the following cases.

Case 1: and . This occurs with probability .

Case 2: , , , and . In this case, there must be some such that . Since , we must have . For each ,

Therefore, we can write

where the last inequality follows since , , , and is large enough.

Putting the two cases together, we have:

Finally, by linearity of expectation, we conclude that

proving the lemma. ∎

The proof of Lemma 4.5 is quite technical and will be given in §5. It is the only part of our argument that relies on the fact that is -free. To explain why proving Lemma 4.5 is difficult, consider an arbitrary color . The value depends on which of the neighbors of in are kept. This, in turn, is determined by what happens to the neighbors of the neighbors of . Since we are only assuming that is -free, the neighborhoods of the neighbors of can overlap with each other. Roughly speaking, we will need to carefully analyze the structure of these overlaps to make sure that Talagrand’s inequality can be applied.

5. Proof of Lemma 4.5

Throughout this section, we shall use the following parameters, where is given in the statement of Lemma 3.1:

Fix a vertex and a color . We need too show that, with high probability, the random variable does not significantly exceed its expectation. To this end, we make the following definitions:

Then . We will show that is highly concentrated and prove that, with high probability, is not much lower than its expected value. Using the identity will then give us the desired upper bound on .

Lemma 5.1 ().

.

Proof.

We use Theorem 2.4, Exceptional Talagrand’s Inequality. Let . In other words, is the set of neighbors of whose lists include a color corresponding to . Then the set is determined by the coloring outcomes of the vertices in . More precisely, as in the proof of Lemma 4.3, we arbitrarily order the colors in for each and let be the random variable that is equal to if and if and is the -th color in . Then , is a list of independent trials whose outcomes determine . Let be the set of outcomes of these trials. Let and define to be the set of all outcomes in which there is a color such that . We claim that satisfies conditions 1 and 2 of Theorem 2.4 with and .

To verify 1, take and event . Each vertex satisfies or there exists such that , and . We call such a conflicting neighbor of . Form a subset of trials by including, for each , the trial itself and, if applicable, the trial corresponding to any one conflicting neighbor of . Since , we have . Now suppose that satisfies . For each vertex , the outcomes of either the trial or the trial of a conflicting neighbor of must be different in and in . Since , every can be a conflicting neighbor of at most vertices . Therefore, and must differ on at least trials, as desired.

It remains to show , where . For any , the number of colors in is a binomial random variable with at most trials, each having probability . Let denote this random variable. Note that . By the union bound, we have

where the last inequality holds for large enough. By the union bound and the fact that (by the assumptions of Lemma 3.1), we get