Counting colorings of triangle-free graphs

By a theorem of Johansson, every triangle-free graph G of maximum degree Δ has chromatic number at most (C+o(1))Δ/logΔ for some universal constant C > 0. Using the entropy compression method, Molloy proved that one can in fact take C = 1. Here we show that for every q ≥ (1 + o(1))Δ/logΔ, the number c(G,q) of proper q-colorings of G satisfies c(G, q) ≥ (1 - 1/q)^m ((1-o(1))q)^n, where n = |V(G)| and m = |E(G)|. Except for the o(1) term, this lower bound is best possible as witnessed by random Δ-regular graphs. When q = (1 + o(1)) Δ/logΔ, our result yields the inequality c(G,q) ≥ exp((1 - o(1)) logΔ/2 n), which implies the optimal lower bound on the number of independent sets in G due to Davies, Jenssen, Perkins, and Roberts. An important ingredient in our proof is the counting method that was recently developed by Rosenfeld. As a byproduct, we obtain an alternative proof of Molloy's bound χ(G) ≤ (1 + o(1))Δ/logΔ using Rosenfeld's method in place of entropy compression.

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1. Introduction

1.1. Counting colorings

All graphs in this paper are finite, undirected, and simple. A celebrated theorem of Johansson [Johansson] says that every triangle-free graph of maximum degree satisfies for some universal constant . (Here and throughout the paper, indicates a function of that approaches as .) The best currently known value for the constant is given by the following result of Molloy:

Theorem 1.1 (Molloy [Molloy]).

If is a triangle-free graph of maximum degree , then

In this paper we establish a lower bound on the number of proper -colorings of when (i.e., when is -colorable by Theorem 1.1). Here is our main result:

Theorem 1.2 ().

For each , there is such that the following holds. Let be a triangle-free graph of maximum degree at most . Then, for every , we have

(1.1)

where , , and .

It was shown by Csikvári and Lin [CL_Sidorenko, Corollary 1.2] that if

is bipartite, i.e., has no odd cycles, then

for all (this is a special case of the so-called Sidorenko conjecture on the number of homomorphisms from a bipartite graph to a fixed graph [Sidorenko]). Our result asserts that approximately the same lower bound holds for triangle-free graphs , under the assumption that .

The bound in Theorem 1.2 has a natural probabilistic interpretation. Suppose is a graph with vertices and edges. If we assign a color from to each vertex of

independently and uniformly at random, what is the probability

that the resulting -coloring is proper? This problem is equivalent to computing since . For each edge , let be the random event that the endpoints of get distinct colors. Then , so if the events were mutually independent, we would have

Theorem 1.2 says that when is triangle-free and , the actual value of is not too much smaller than this “naive” bound. Notice that, since ,

which enables us to treat the factor in (1.1) as an error term. (On the other hand, below the threshold, i.e., for , the value tends to infinity as a function of .)

Let us now explore some of the consequences that can be derived from Theorem 1.2 by applying it to specific values of . By substituting for , we get the following:

Corollary 1.3 ().

The following holds for each . Let be an -vertex triangle-free graph of maximum degree at most . If , then

Proof.

A direct calculation using (1.1) and the bounds and . ∎

As an immediate consequence of Corollary 1.3, we obtain the optimal lower bound on the number of independent sets in triangle-free graphs due to Davies, Jenssen, Perkins, Roberts:

Corollary 1.4 (Davies–Jenssen–Perkins–Roberts [indep, Theorem 2]).

Let be an -vertex triangle-free graph of maximum degree at most . Then

where denotes the number of independent sets in .

Proof.

Fix any and set . Since a proper -coloring of is a sequence of independent sets in that partition , we have . Therefore, by Corollary 1.3,

As can be taken arbitrarily small, the desired result follows. ∎

Theorem 1.2 is also interesting for larger values of . For instance, we can take :

Corollary 1.5 ().

Let be an -vertex triangle-free graph of maximum degree at most . Then

Proof.

Follows by substituting for in (1.1) and using the bound . ∎

Even though every graph of maximum degree is -colorable, the conclusion of Corollary 1.5 may fail for graphs that are not triangle-free. For instance, if is a disjoint union of cliques of size , then, using Stirling’s formula, we obtain

which is less than the bound in Corollary 1.5 roughly by a factor of .

It is natural to wonder haw sharp our lower bound on is. We show that it is optimal (modulo the error term ):

Theorem 1.6 ().

Fix positive integers and . For every sufficiently large such that is even, there exists a triangle-free -regular graph with

(1.2)

where , , and .

We prove Theorem 1.6 in §4 by showing that the bound (1.2) holds for the random -regular graph with high probability.

1.2. Counting DP-colorings

Molloy proved his Theorem 1.1 not just for the ordinary chromatic number , but also for the list-chromatic number . In fact, as shown in [Bernshteyn], the same upper bound holds in the more general setting of DP-coloring (also known as correspondence coloring), introduced by Dvořák and Postle [DP]. Recall that in the context of list-coloring, each vertex of a graph is given its own list of colors to choose from, and the goal is to find a proper -coloring of , i.e., a mapping such that for all and whenever . (Ordinary coloring is a special case of this when all lists are the same.) DP-coloring further generalizes list-coloring by allowing the identifications between the colors in the lists to vary from edge to edge. Formally, DP-coloring is defined using an auxiliary graph called a DP-cover:

Definition 1.7.

A DP-cover of a graph is a pair , where is a graph and is an assignment of subsets to the vertices satisfying the following conditions:

  • The family of sets is a partition of .

  • For each , is an independent set in .

  • For , , the edges of between and form a matching; this matching is empty whenever

We call the vertices of 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 for all . A (partial) -coloring is proper if the image of is an independent set in , i.e., if for all , .

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 .

To see that list-coloring is a special case of DP-coloring, consider the following construction. Suppose that each vertex of a graph is given a list of colors to choose from. Define

(thus, the sets for different vertices are disjoint) and let be the graph with vertex set

in which vertices and are adjacent if and only if and . Then is a DP-cover of and there is a natural one-to-one correspondence between the proper -colorings and the proper -colorings of .

We prove the following generalization of Theorem 1.2:

Theorem 1.8 ().

For each , there is such that the following holds. Let be a triangle-free graph of maximum degree at most . Then, for all and every -fold DP-cover of , the number of proper -colorings of is at least

where , , and .

The problem of counting DP-colorings was studied by Kaul and Mudrock in [Kaul], where they introduced the DP-color function . By definition, is the minimum number of proper -colorings of taken over all -fold covers of . Using this terminology, we can say that Theorem 1.8 provides a lower bound on for triangle-free graphs of maximum degree when .

An interesting feature of the lower bound given by Theorem 1.8 is that it is sharp (modulo the error term ) for every graph , as was shown by Kaul and Mudrock:

Theorem 1.9 (Kaul–Mudrock [Kaul, Proposition 16]).

For every graph with vertices and edges and every , there is a -fold DP-cover of such that the number of proper -colorings of is at most .

1.3. Overview of the proof

In this subsection we outline the key ideas that go into the proofs of our main results. For simplicity, we shall focus on Theorem 1.2; the more general argument needed to establish Theorem 1.8 in the DP-coloring setting is virtually the same, except for a few minor technical changes.

Let be a triangle-free graph of maximum degree at most and let . Our approach is inspired by Molloy’s proof of the bound (i.e., of Theorem 1.1). To explain Molloy’s strategy, we need to introduce some notation and terminology. Let be a proper partial -coloring of . For each vertex , we let be the set of all colors such that no neighbor of is colored . Also, for , let be the number of uncolored neighbors of such that . Define the following numerical parameters:

The partial coloring is good if it satisfies the following two conditions:

  • for every uncolored vertex , ; and

  • for every uncolored vertex and , .

In order to find a proper -coloring of , Molloy establishes two auxiliary results:

  1. [label=(M0)]

  2. admits a good proper partial -coloring.

  3. Every good partial coloring can be extended to a proper -coloring of the entire graph .

Statement 2 is proved using the Lovász Local Lemma and is by now standard (its first appearance is in the paper [Reed] by Reed; see also [MolloyReed, §4.1] for a textbook treatment). On the other hand, Molloy’s proof of 1 was highly original and combined several novel ideas. In particular, it relied on a technique introduced by Moser and Tardos in [MT] and called the entropy compression method (the name is due to Tao [Tao]). Initially designed as a means to establish an algorithmic version of the Lovász Local Lemma, entropy compression has by now become an invaluable tool in the study of graph coloring; see, e.g., [Esperet, BCGR, Duj] for a sample of its applications. An alternative approach—with the so-called Lopsided Lovász Local Lemma taking the place of entropy compression—was developed by the first named author in [Bernshteyn]. The ideas of [Molloy] and [Bernshteyn] have been pursued further by a number of researchers in order to strengthen and extend Theorem 1.1 in various ways [fraction, locallistsize, DKPS].

Very recently, Rosenfeld [Rosenfeld] discovered a remarkably simple new technique that can be used as a substitute for entropy compression. A number of applications of Rosenfeld’s method to (hyper)graph coloring appear in the paper [WW] by Wanless and Wood, which also describes a general framework for applying Rosenfeld’s technique to coloring problems. One benefit of Rosenfeld’s approach (in addition to its simplicity) is that it not only proves the existence of an object with certain properties (such as a coloring), but also gives a lower bound on the number of such objects. This makes it particularly well-suited for our purposes. As a byproduct of our proof of Theorem 1.2, we obtain a new simple proof of 1 (and hence of Molloy’s Theorem 1.1) using Rosenfeld’s technique in lieu of entropy compression or the Lopsided Lovász Local Lemma. A different proof of Molloy’s theorem using Rosenfeld’s method was given in [Pirot] by Pirot and Hurley (their work was carried out independently from ours).

Let us now describe the main steps in our argument.

  1. [wide]

  2. Besides the use of the entropy compression method, Molloy’s proof of 1 involved another novel ingredient, namely a version of the coupon-collector theorem for elements drawn uniformly at random from sets of varying sizes [Molloy, Lemma 7]. Our proof uses this result as well. In fact, we need a slightly stronger version of it, because in our setting may be significantly larger than and because we need the error bounds to be more precise. We state and prove this strengthening in §3.2.

  3. Next, in §3.3, we give a lower bound on the number of proper partial colorings of . This is done via an analysis of the greedy coloring algorithm. That is, we color the vertices of one by one, where each next vertex is either left uncolored or assigned an arbitrary color that has not yet been used by any of its neighbors. Using the coupon-collector result from §3.2, we argue that, on average, each vertex will have many available colors to choose from, which yields the desired lower bound on the total number of proper partial colorings. The bound we obtain here is already sufficient to deduce the lower bound on the number of independent sets in given by Corollary 1.4.

  4. In §3.4 we use a version of Rosenfeld’s method to argue that a fairly large fraction of all proper partial colorings of are good (in particular, a good coloring exists). Combined with the result in §3.3, this yields a lower bound on the number of good colorings.

  5. As mentioned earlier, a simple application of the Lovász Local Lemma shows that every good partial coloring can be extended to a proper coloring of . We need to know not only that such an extension exists, but also how many such extensions there are. Thankfully, the Lovász Local Lemma can be used to derive an explicit lower bound on the probability that a random extension of is proper, which can be translated into a lower bound on the number of such extensions. This is accomplished in §3.5.

  6. Finally, in §3.6, we combine all the above results to derive a lower bound on the number of proper colorings of . Some care has to be taken because the same proper coloring of may arise as an extension of several good partial colorings. Nevertheless, we are able to use a double counting argument to account for this and obtain the desired result. Curiously, the double counting at this stage is the main contributor to the error term in the statement of Theorems 1.2 and 1.8.

2. Probabilistic preliminaries

The following is a standard form of the Chernoff inequality:

Lemma 2.1 ([McDiarmid1998, Theorem 2.3(b)]).

Suppose that , …,

are independent random variables with

for each . Let . Then, for any ,

We also need a version of the Chernoff bound for negatively correlated random variables, introduced by Panconesi and Srinivasan [panconesi]. We say that -valued random variables , …, are negatively correlated if for all ,

Lemma 2.2 ([panconesi, Theorem 3.2], [Molloy, Lemma 3]).

Let , …, be -valued random variables. Set and . If , …, are negatively correlated, then

We shall use the Lovász Local Lemma in the following quantitative form:

Lemma 2.3 ([prob_method, Lemma 5.1.1]).

Let be a finite set of random events. For each , let be a subset of such that is mutually independent from the events in . If there exists an assignment of reals to the events such that

then the probability that no event in happens is at least .

More specifically, we will need the following consequence of Lemma 2.3:

Corollary 2.4 (Quantitative Symmetric Lovász Local Lemma).

Let be a finite set of random events. For each , let be a subset of such that is mutually independent from the events in . Suppose that for all , and , where and . If , then

Proof.

Take in the statement of Lemma 2.3. ∎

3. Proof of Theorem 1.8

3.1. Standing assumptions and notation

Throughout §3, we fix the following data:

  • a real number ;

  • an integer , assumed to be large enough as a function of ;

  • an integer satisfying ;

  • a triangle-free graph of maximum degree at most with vertices and edges;

  • a -fold DP-cover of .

As mentioned in §1.3, we also define

The neighborhood of a vertex is the set of all neighbors of in . The closed neighborhood of is the set , and the second neighborhood is the set of all vertices at distance at most from . For a subset , we write and . Given , the notation , , etc. is defined analogously but with respect to the graph instead of . For a set and a vertex , we use to denote the set .

When is a partial function and is undefined for some element , we write . Given a partial -coloring of and , we let

Also, for each , we let

(Recall that here is the underlying vertex of the color .)

3.2. A coupon-collector lemma

In this subsection, we establish a version of the coupon-collector theorem that slightly generalizes [Molloy, Lemma 7] by Molloy. Our argument closely follows Molloy’s proof.

Lemma 3.1 (Coupon-collector).

Let , , …, be finite sets, where and (there are no assumptions on for ). For each , let be a matching between and . For every , pick an element uniformly at random from , making the choices for different independently. This defines a random partial function on the set . Let

and, for each , let

Then the following statements are valid:

  1. [label=()]

  2. .

  3. .

  4. .

Proof.

Without loss of generality, we may assume that for all . For each , let be the set of all indices such that is matched to some . Define a quantity by

Observe that, since each is a matching,

(3.1)

Notice also that, since , for large enough we have

(3.2)

1 Using the inequality valid for all , we obtain

(3.3)

Applying Jensen’s inequality to (3.3) and using (3.1), we get

as desired. Note that, since , we also have .

2 For , let

be the indicator random variable of the event

and let . We claim that the random variables are negatively correlated:

Claim 3.1.a.

For any , .

Proof of Claim 3.1.a.

We first notice that for any and ,

(3.4)

To see this, for each and , let be the set of all elements such that (so contains at most one element). To sample conditioned on the event , we pick each uniformly at random from . As for all and , the removal of from does not decrease the probability that for each , there is with , so (3.4) holds. Next, we observe that (3.4) is equivalent to

(3.5)

Applying (3.5) inductively establishes the claim. ∎

Recalling that , using Lemma 2.2, and invoking (3.2), we obtain

3 Consider any . If , then, using the inequality valid for all , we can write

(3.6)

If, on the other hand, , then

Since the values for distinct are chosen independently, we may apply Lemma 2.1 to get

for any . We may assume (otherwise with probability ) and plug in the value , which yields

(3.7)

Since , it follows from (3.6) and (3.7) that for all ,

Therefore, we may conclude that

assuming is large enough. ∎

3.3. Counting partial colorings

Let denote the set of all proper partial -colorings of . Also, for a subset , let be the set of all proper partial -colorings with . In this subsection we establish a lower bound on . We start with a lemma:

Lemma 3.2 ().

Suppose that and . Then

Proof.

To begin with, observe that

(3.8)

since given a partial coloring , we can extend it to by assigning to an arbitrary color from . To get a lower bound on the right-hand side of (3.8), we shall use Lemma 3.1. For a proper partial -coloring , let denote the set of all extensions of to , i.e., all proper partial -colorings that agree with on . Since is triangle-free, a coloring is obtained by assigning to each an arbitrary color from . Therefore, we may apply Lemma 3.11 with and the sets and playing the role of , , …, to conclude that

Now we can write

Combining this with (3.8) yields the desired result. ∎

Corollary 3.3 (Counting partial colorings).

We have

Proof.

Let , …, be an arbitrary ordering of the vertices of . Since, by definition, , repeated applications of Lemma 3.2 yield

As mentioned in the introduction, Corollary 3.3 can already be used to derive the lower bound on the number of independent sets in given by Corollary 1.4.

3.4. Counting good partial colorings

Let be a proper partial -coloring of . We say that has a flaw at a vertex if and at least one of the following holds:

  • , or

  • for some .

Let be the set of all vertices such that has a flaw at . If , we say that is good. The set of all good partial -colorings of is denoted by . Our goal in this subsection is to establish a lower bound on .

Given a subset , we say that a partial -coloring is good on if for every vertex such that . Let denote the set of all that are good on . We emphasize that a coloring is not required to belong to , i.e., the domain of may not be a subset of . However, whether or not is good on only depends on the restriction of to (because whether or not has a flaw at is determined by the restriction of to ). Since every proper partial -coloring is vacuously good on the empty set, we have

Lemma 3.4 ().

Suppose that and . Then

(3.9)
Proof.

This is an inductive argument in the style of Rosenfeld [Rosenfeld]. Note, however, that our application of Rosenfeld’s method is somewhat different from the ones in [Rosenfeld, WW]. Namely, we do not show that grows by a certain factor compared to , but rather that it does not shrink too much. This difference appears crucial for our approach. We remark that in [Pirot], Pirot and Hurley prove Molloy’s bound using a more “standard” version of Rosenfeld’s technique (their argument does not refer to good partial colorings at all).

We proceed by induction on . So, fix and suppose that (3.9) holds when is replaced by any set of strictly smaller cardinality. Let be the set of all such that is good on but not on . Then

For each , define . If , then there must be a vertex such that . Since , this implies that , and hence

We will give an upper bound for for each .

Claim 3.4.a.

Set and . Then, for every ,

Proof of Claim 3.4.a.

Let be the set of all proper partial -colorings that are good on such that . For each , let be the set of all extensions of , i.e., all proper partial -colorings of that agree with on . Also, let be the set of all that have a flaw at . Since is triangle-free, a coloring is obtained by assigning to each vertex an arbitrary color from . Thus, we may use parts 2 and 3 of Lemma 3.1 with and the sets and playing the role of , , …, to conclude that

Note that if and