Colouring Graphs with Sparse Neighbourhoods:Bounds and Applications

Colouring Graphs with Sparse Neighbourhoods: Bounds and Applications

Abstract

Let be a graph with chromatic number , maximum degree and clique number . Reed’s conjecture states that for all . It was shown by King and Reed that, provided is large enough, the conjecture holds for . In this article, we show that the same statement holds for , thus making a significant step towards Reed’s conjecture. We derive this result from a general technique to bound the chromatic number of a graph where no vertex has many edges in its neighbourhood. Our improvements to this method also lead to improved bounds on the strong chromatic index of general graphs. We prove that provided is large enough.

1 Introduction

It is well known that the chromatic number of a graph is bounded above by , where denotes the maximum degree of . Similarly, a trivial lower bound on is given by the clique number , which is the largest number of pairwise adjacent vertices in . In 1998, Reed conjectured that, up to rounding, the chromatic number of a graph is at most the average of these two bounds.

Conjecture 1.1.

[13] If is a graph, then .

As evidence for his conjecture, Reed proved that the chromatic number can be bounded above by a non-trivial convex combination of and .

Theorem 1.2.

[13] There exists such that for every graph , we have .

King and Reed [10] subsequently gave a shorter proof of Theorem 1.2 by exploiting a recent result of King [9] on independent sets hitting every maximal clique. Using King’s result, it suffices to prove Theorem 1.2 for graphs with clique number . Given this fact, there are two main steps in the proof of King and Reed. The first is to show that if such a graph is also critical, then no neighbourhood contains many edges. More precisely, there exists such that every neighbourhood induces at most edges. We say that such a graph is -sparse. The second step is to invoke the naive colouring procedure and the probabilistic method to colour the graph. Indeed, using these techniques, it can be shown that a -sparse graph is -colourable for some depending on . This completes the proof.

Seeking only a short proof of Theorem 1.2, King and Reed did not optimise the two steps of their method. Approximately, they find that and suffice. However, since Reed’s Conjecture is equivalent to proving Theorem 1.2 for , it is natural to ask if one can increase the value of obtained. It would suffice to provide an improved answer to any of the two following questions. Recall first that a graph is -critical if is not -colorable but every proper subgraph of is.

Question 1.3.

Let be a -critical graph with . What is the largest , such that is -sparse?

Question 1.4.

Let be a -sparse graph. What is the largest such that ?

1.1 Main Results

In this paper we improve on the best known results for both of these questions. In fact, we prove results in the context of list colouring, a generalization of colouring. A list assignment is a function that to each vertex assigns a nonempty set of colours. An -colouring is a coloring of such that for every . A -list-assignment is a list assignment such that for every . A graph is -list-colourable if has an -coloring for every -list-assignment . The list chromatic number of , denoted is the minimum such that is -list-colourable. We say a graph is -critical with respect to a list assignment if does not have an -colouring but every proper subgraph of does.

In response to Question 1.3, we prove the following theorem.

Theorem 1.5.

Let such that . If is -critical with respect to some -list-assignment and , then is -sparse.

Note that this implies the same result for -critical graphs. King and Reed [10] showed that if is a -critical graph with clique number , then is -sparse provided . Setting in Theorem 1.5, our bound gives , an eightfold improvement.

Question 1.4 is a well studied problem. Molloy and Reed [11] proved that, for , one may take provided that the maximum degree is large enough. More recently, with the same conditions, Bruhn and Joos [2] improved this to . These bounds are approximations of more complicated expressions, see [11] and [2] respectively. Both of these results are proved using a single application of the naive colouring procedure, a randomised colouring technique which generates a partial proper colouring of a -sparse graph. In this article, we develop an iterative version and using this we improve the bound of Bruhn and Joos by a factor of as follows.

Theorem 1.6.

Let be a -sparse graph with , and let . There exists such that if , then .

In fact, we prove Theorem 1.6 in the setting of correspondence colouring defined in Section 3, a generalization of list colouring. The use of correspondence colouring allows us to simplify some of the intricacies in the proof and is quite natural in this setting.

This paper is not the first to consider an iterative application of the naive colouring procedure. Indeed, the notable result of Johansson [7], which states that triangle-free graphs satisfy is proved in this way, see also [12]. Triangle-free graphs behave particularly nicely with respect to an iterative version because, for any partial colouring, the subgraph induced by the uncoloured vertices is still triangle-free. We should briefly remark however that the method of Johannson [7] is somewhat different in the sense that the procedure is only applied to a fraction of the vertices in each step. In this case the technique is often called the semi random method or Rödl nibble and can be traced back to [1, 15].

In this paper, we show that for -sparse graphs, the naive colouring procedure can generate a partial colouring with the additional property that the uncoloured subgraph is almost -sparse (see Lemma 3.20). This is the key which allows us to apply the procedure iteratively to the uncoloured subgraph. In addition, the probability that a vertex remains coloured is about (see Proposition 3.7) and hence the probability a vertex is in is about . After one iteration, Bruhn and Joos had shown that the difference between the maximum degree of and the resulting list sizes had decreased by at least ; if that was the initial difference, then we could greedily colour to finish. However, given the key lemma that is almost -sparse, we may apply the procedure again. In each step, we accrue a new savings proportional to the current maximum degree. Terminating this procedure ad infinitum would result in roughly the following savings:

Of course, we cannot carry out this iteration indefinitely, but after four iterations, we have saved as much as claimed in Theorem 1.6. For technical reasons, we adopt a different perspective in the proof of Theorem 1.6, wherein we study the ratio of maximum degree to list size and show that as long as this ratio is at most that of Theorem 1.6, then the ratio will slowly decrease after each iteration until it falls below whereupon we finish by colouring greedily.

By using Theorem 1.5 and Theorem 1.6 together with the technique of King and Reed, we obtain that the -version of Reed’s Conjecture holds for .

Theorem 1.7.

There exists such that if is a graph of maximum degree and clique number , then .

1.2 The Strong Chromatic Index

The strong chromatic index, , of a graph is defined as the least integer for which there exists a -colouring of such that edges at distance at most receive different colours. Equivalently, , where denotes the square of the line graph of . Since , the trivial upper bound on the chromatic number gives that . However Erdős and Nešetřil conjectured a much stronger upper bound, see [6].

Conjecture 1.8.

If is a graph, then .

If true, this bound would be tight. Indeed, if denotes the graph obtained from a -cycle by blowing up each vertex into vertices, then and is a clique with vertices. Figure 1 depicts the graph .

Figure 1: A blow-up of the -cycle.

In 1997, Molloy and Reed made the first step towards Conjecture 1.8. They showed that for all graphs , the graph is a subgraph of a graph such that and is -sparse. Thus the naive colouring procedure guarantees that (and hence ) can be coloured with colours for some .

Theorem 1.9.

[11] There exists such that if is a graph with sufficiently large maximum degree , then .

With and their colouring procedure, the value of that Molloy and Reed obtain is approximately . Bruhn and Joos [2] improved the bound on the neighbourhood sparsity and showed that is asymptotically -sparse. With , say, and their colouring procedure, they deduce Theorem 1.9 for . This gives the following.

Theorem 1.10.

[2] If is a graph of sufficiently large maximum degree , then .

In this article we improve the bound in Theorem 1.10. To do this we first show that one only needs to colour a subgraph of consisting of high degree vertices with many neighbours of high degree. This idea resembles the notion that one need only colour a critical subgraph of . We then show that admits a much better bound on its neighbourhood sparsity than . Combined with Theorem 1.6, we obtain the following result.

Theorem 1.11.

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

1.3 Outline of the Paper

In Section 2 we deal with Question 1.3 and prove Theorem 1.5. In Section 3 we address Question 1.4. We recall the naive colouring procedure and develop an iterative version. We then derive Theorem 1.6 as a consequence. Section 4 is devoted to the strong chromatic index and the proof of Theorem 1.11. Finally, in Section 5, we prove Theorem 1.7.

For standard definitions and graph theoretic notation, we refer the reader to Diestel [3].

2 A Density Lemma

In this section we prove Theorem 1.5, which guarantees that a graph that is critical with respect to some -list-assignment is -sparse, for some depending on and the clique number of . To do this, we first show that if is an -critical graph with respect to some -list-assignment , then the minimum degree of an induced subgraph of cannot be too large.

Proposition 2.1.

If is an -critical graph with respect to some -list-assignment , then for all induced subgraphs of , we have .

Proof.

Suppose for a contradiction that is an induced subgraph of with . Let and note that for every vertex , we have . Since is -critical, has an -colouring . Now to each vertex , assign a list of colours , defined by . For each , we have . Hence can be extended to an -colouring of , a contradiction. ∎

The bound in Proposition 2.1 exhibits an awkward dependence on , and so we first derive an upper bound on this parameter. Note that we let denote the complement of . One can easily guarantee a large matching in the complement of a graph if the clique number is small.

Proposition 2.2.

For every graph , has a matching of size at least .

Proof.

If is a maximal matching in , then is a clique. Thus . ∎

We make use of the following classical result of Erdős, Rubin and Taylor [5].

Theorem 2.3.

[5] Let be an integer. If is a complete -partite graph where each partition class contains at most two vertices, then .

Using Proposition 2.2 and Theorem 2.3, we can derive the desired bound.

Proposition 2.4.

If is a graph then .

Proof.

By Proposition 2.2, the graph has an antimatching of size . Let denote the graph with vertex set and edge set . Note that satisfies the conditions in Theorem 2.3 with . Thus, by Theorem 2.3, we have . Now, since , we have as desired. ∎

Let be a graph and be a subset of with . We say that is a minimum-degree ordering of if is a vertex of minimum degree in the subgraph , for all . We use this ordering to derive a first bound on .

Lemma 2.5.

Let be a graph of maximum degree and clique number . If is -critical with respect to some -list-assignment , then for every vertex , we have

Proof.

Let be a vertex of with , and let . Also, let denote the graph formed from by adding independent vertices. We do this so as to compare more easily with , as it is the maximum number of edges in the neighborhood of a vertex of degree . Finally, let be a minimum-degree ordering of , and set for . Clearly, we have . For , the vertex is isolated, and thus . On the other hand, for , the vertex has degree by Proposition 2.1, so we have

By Proposition 2.4, we have . Furthermore, for each . Thus, we have:

(1)

The second term in the maximum of (1) eventually becomes negative when . Because of the maximum, we may truncate the sum and deduce that

We can now prove Theorem 1.5.

Proof of Theorem 1.5.

Let . By Lemma 2.5, we have for every vertex that

Hence is -sparse. ∎

3 A Sparsity Lemma

3.1 The Naive Colouring Procedure

The naive colouring procedure is a well studied technique which generates a partial proper -colouring of a graph . In the context of graph colourings it was first used by Kahn [8], though it had already appeared in a more abstract setting [1]. We refer the reader to [12] for a survey on further applications of the technique. In its simplest form, the naive colouring procedure consists of the following two steps.

  1. To each vertex , assign a colour chosen uniformly at random from .

  2. If and are adjacent vertices with the same colour, then uncolour both and .

Let be a graph and be an integer with . If no vertex of has too many edges in its neighbourhood, then one can show that with positive probability, the partial -colouring generated by the above procedure has the property that vertices of large degree see many repeated colours in their neighbourhoods. To be more precise, let denote the number of coloured vertices in and let denote the number of distinct colours amongst the colours of the vertices in . If there are repeated colours in , then clearly . The following proposition states that if the difference is large enough, then such a partial colouring can be extended to a colouring of the whole graph in an efficient way.

Proposition 3.1.

Let be a graph and be an integer such that . If there is a partial proper -colouring of such that for every vertex , we have , then has a -colouring.

Proof.

Let be an uncoloured vertex. The number of uncoloured neighbours of is precisely . The number of colours in which do not appear in is . It remains to list colour the uncoloured subgraph , where every vertex has a list of size at least one greater than . Such a colouring can be constructed greedily. ∎

It is hard to analyse the expectation of the random variable . However, by inclusion-exclusion, it is easy to see that , where and denote the number of pairs and triples of vertices in which are all coloured the same and all remain coloured after the procedure. When computing the expectation of and , it is convenient to assume that the graph in question is -regular. Indeed, this is no restriction, since if is a graph of maximum degree , then may be embedded in a -regular graph by iterating the following process. Take two copies of and add edges between corresponding vertices of degree less than . Note that and if is -sparse, then so is . In this way, we will frequently assume that the graph under consideration is -regular.

Once the expectations of and have been calculated, one can show that they are concentrated about their expectations. In other words, the probability that is far from its expectation is very small. The Lovász Local Lemma can then be applied to ensure that this is the case for every .

Lovász Local Lemma.

Let , a postive integer, and a finite set of (bad) events such that for every ,

  • , and

  • There exists a set of events of size at most such that is mutually independent of .

If , then there exists an outcome in which none of the events in occur.

In this paper we show that the naive colouring procedure can be iterated. More precisely, we prove that if is a -sparse graph, then after a single application of the procedure the graph induced by the uncoloured vertices retains some of the sparsity of the original graph. Thus we can apply the procedure again to the uncoloured subgraph. In order to show that the sparsity is retained, we first show that with positive probability, the set of uncoloured vertices behaves somewhat randomly. The precise condition that we require is the following.

Definition 3.2.

Let , be a graph with maximum degree , and . We say that is a -quasirandom subgraph of if for every pair of not necessarily distinct vertices , we have

Note that for , the condition in Definition 3.2 reduces to . To show that the uncoloured subgraph is a -quasirandom subgraph of , we track more random variables which count the number of uncoloured vertices in the common neighbourhood of two vertices. These random variables will also be shown to be highly concentrated, and so we can add the corresponding bad events to our previous application of the Lovász Local Lemma.

3.2 Correspondence Colouring

Any iterative application of the naive colouring procedure necessitates the introduction of lists of colours. This is because in each step, some colours are forbidden at a vertex , namely those which have been assigned to the neighbours of in a previous application. In analysing the procedure, a technical issue arises due to the fact that the probability a vertex keeps a particular colour in its list may vary depending on the vertex and the colour. Previously, this issue has been dealt with by introducing extra vertices, or coin flips, to equalise the probabilities.

Here, we use a generalisation of list colouring called correspondence colouring, introduced by Dvořák and the third author in [4] (and sometimes referred to as DP-coloring). As well as proving a more general statement, the use of correspondence coloring automatically equalises the probabilities, and thus simplifies the proof. Here is the definition we use which is equivalent to but slightly different from the definitions given elsewhere.

Definition 3.3.

[4] Let be a graph, and let be an arbitrary orientation of .

  • A correspondence assignment of is a function defined on as follows: To each vertex , assigns a set , and to each edge , assigns an injective partial function such that for every edge .

  • If each has size at least , then is a -correspondence assignment for .

  • A -colouring of is a function such that for every , and for every edge , either or .

  • The correspondence chromatic number of , denoted , is the smallest integer such that is -colourable for every -correspondence assignment .

We say that the function assigned to the edge is total if . Note that there is no requirement that functions in the definition above are total. Hence the following definition.

Definition 3.4.

Let be a graph and be a correspondence assignment of . We say is total if and are total for every edge of .

Note that if is total and is connected, then for every pair of vertices . We remark if is a correspondence assignment of a graph such that for every pair of vertices , then we will often extend to a total correspondence assignment by arbitrarily extending each function to be total. Clearly, if is -colourable, then is also -colourable.

Definition 3.5.

Let be a graph and let be a total correspondence assignment of . If , , , then we say and correspond under if , or equivalently, . If the correspondence assignment is clear from the context, then we simply say that and correspond.

Note that Proposition 3.1 is still valid for correspondence colouring.

We now state precisely the variant of the naive colouring procedure that we use. Let be a -correspondence assignment.

Procedure 3.6.

Suppose is a graph and is a correspondence assignment for . We generate a partial -colouring as follows.

  1. Assign each vertex a colour chosen uniformly at random from .

  2. For every edge , pick an end uniformly at random, that is with probability and with probability .

  3. For each vertex , let if and only if for every edge , at least one of the following hold: or . (Equivalently, uncolour if there exists an edge such that and .)

We remark that the uncolouring method used here in Steps 2 and 3 was also used by Bruhn and Joos [2]. Before analysing the procedure, we note the following fundamental fact.

Proposition 3.7.

Let be a -regular graph and let be a total -correspondence assignment of . For every vertex , the probability that is coloured after an application of Procedure 3.6 (that is ) is .

Proof.

Let be the event that . For each neighbour of , let be the event that and . Now by definition, . Since these events are independent, we find that

Note that since the events are independent. Since all correspondences are total, . Furthermore, . Hence and .

Thus . As is -regular, . Hence as desired. ∎

We are ready to prove the key lemma of this section. The result is similar to Lemma 7 in Bruhn and Joos [2], however we extend it to correspondence colouring, and we ensure that the uncoloured vertices induce a -quasirandom subgraph.

Lemma 3.8.

Let be a -regular -sparse graph and let be a -correspondence assignment for . Also let satisfy

There exists an integer such that if , then there is a -quasirandom subgraph of , and a -correspondence assignment of such that any -colouring of extends to a -colouring of , where and .

Proof of Lemma 3.8.

We may assume that for each vertex , the set has size precisely (by restricting to an arbitrary subset of of size ). Furthermore, we may assume that is a total correspondence assignment (by extending, for each edge , the function to an arbitrary total function and setting ). Note the latter two assumptions only restrict the possible set of -colourings.

Now consider an application of Procedure 3.6 to the graph , which produces a partial -colouring of . Let be the subgraph of induced by the uncoloured vertices, and let be the correspondence assignment obtained from as follows: For each , let . To every edge in , let assign the map , where is the restriction of to and .

We set . Note that every -colouring of can be extended to a -colouring of by letting if and otherwise. Moreover, we could truncate each to an arbitrary subset of size restricting further the possible -colorings. However, this is not technically needed since the definition of -correspondence assignment we use requires only lists of size at least (not necessarily equal to) .

It remains to show that both of the following hold with high probability: is -quasirandom; and .

To this end we define a collection of events and random variables. Firstly, for each pair of vertices such that the distance from to is at most , we define a random variable by . In particular, for a vertex , we have . Let be the event that

We show that the probabilities of all these bad events are small in the following two claims.

Claim 3.9.

For every , we have

Proof. By Proposition 3.7, we have . Thus . In Section 3.3 we argue that the random variable is highly concentrated about its expectation. More precisely, it follows from Lemma 3.19 that

Hence the conclusion.

For every vertex , let

That is, denotes the number of non-adjacent pairs of vertices in whose colours under correspond to the same colour at . For a graph , let denote the set of triangles of . For every vertex , let

That is, denotes the number of non-adjacent triples of vertices in whose colours under correspond to the same colour at .

For convenience, for each , let be a fixed constant such that induces precisely edges.

We begin by finding a lower bound on the expectation of as follows.

Claim 3.10.

For each vertex , we have

where denotes a function that tends to as tends to infinity.

Proof.

Let . First let and be non-adjacent neighbours of . Let and .

Note that

Yet

since the events are independent.

Thus we proceed to calculate as follows. For each , let be the event that and (that is the event that ‘uncolours’ ). Note for each , and hence .

For each , we have by the union bound that and hence