Defective and Clustered Choosability of Sparse Graphs

06/19/2018 ∙ by Kevin Hendrey, et al. ∙ Monash University 0

An (improper) graph colouring has "defect" d if each monochromatic subgraph has maximum degree at most d, and has "clustering" c if each monochromatic component has at most c vertices. This paper studies defective and clustered list-colourings for graphs with given maximum average degree. We prove that every graph with maximum average degree less than 2d+2/d+2 k is k-choosable with defect d. This improves upon a similar result by Havet and Sereni [J. Graph Theory, 2006]. For clustered choosability of graphs with maximum average degree m, no (1-ϵ)m bound on the number of colours was previously known. The above result with d=1 solves this problem. It implies that every graph with maximum average degree m is 3/4m+1-choosable with clustering 2. This extends a result of Kopreski and Yu [Discrete Math., 2017] to the setting of choosability. We then prove two results about clustered choosability that explore the trade-off between the number of colours and the clustering. In particular, we prove that every graph with maximum average degree m is 7/10m+1-choosable with clustering 9, and is 2/3m+1-choosable with clustering O(m). As an example, the later result implies that every biplanar graph is 8-choosable with bounded clustering. This is the first non-trivial result for the clustered version of the earth-moon problem. The results extend to the setting where we only consider the maximum average degree of subgraphs with at least some number of vertices. Several applications are presented.

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1. Introduction

This paper studies improper colourings of sparse graphs, where sparsity is measured by the following standard definition. The maximum average degree of a graph , denoted by , is the maximum, taken over all subgraphs of , of the average degree of . We consider improper colourings with bounded monochromatic degree or with bounded monochromatic components, for graph classes with bounded maximum average degree. We now formalise these ideas. A colouring of a graph is a function that assigns a colour to each vertex. In a coloured graph , the monochromatic subgraph of is the spanning subgraph consisting of those edges whose endpoints have the same colour. A colouring has defect if the monochromatic subgraph has maximum degree at most ; that is, each vertex is adjacent to at most vertices of the same colour as . A connected component of the monochromatic subgraph is called a monochromatic component. A colouring has clustering if each monochromatic component has at most vertices. Of course, a colouring is proper if and only if it has defect or clustering .

Our focus is on minimising the number of colours, with small defect or small clustering as a secondary goal. This viewpoint leads to the following definitions. The defective chromatic number of a graph class is the minimum integer such that for some integer , every graph in is -colourable with defect . The clustered chromatic number of a graph class is the minimum integer such that for some integer , every graph in is -colourable with clustering .

The above definitions extend in the obvious way to list-colourings and choosability. A list-assignment for a graph is a function that assigns a set of colours to each vertex . A list-assignment is a -list-assignment if for each vertex . An -colouring is a colouring of such that each vertex is assigned a colour in . Define to be -choosable with defect if has an -colouring with defect for every -list-assignment of . Similarly, is -choosable with clustering if has an -colouring with clustering for every -list-assignment of .

Defective and clustered (list-)colouring has been widely studied on a variety of graph classes, including: bounded maximum degree [28, 2], planar [15, 23, 16], bounded genus [15, 3, 14, 43, 13, 25], excluding a minor [21, 24, 41, 39, 35, 38], excluding a topological minor [21, 39], and excluding an immersion [41]. See [42] for a survey on defective and clustered colouring. All of these classes have bounded maximum average degree. Thus our results are more widely applicable than nearly all of the previous results in the field. That said, it should be noted that some of the existing results for more specific graph classes give better bounds on the number of colours or on the defect or clustering. Generally speaking, our results give the best known bounds for graph classes that have bounded maximum average degree, unbounded maximum degree, and have no strongly sub-linear separator theorem. Examples include graphs with given thickness, stack-number or queue-number.

1.1. Defective Choosability

Defective choosability with respect to maximum average degree was previously studied by Havet and Sereni [27], who proved the following theorem.

Theorem 1 ([27]).

For and , every graph with is -choosable with defect .

Our first result improves on Theorem 1 as follows:

Theorem 23).

For and , every graph with is -choosable with defect .

Note that the two theorems are equivalent for . But for , the assumption in Theorem 2 is weaker than the corresponding assumption in Theorem 1, thus Theorem 2 is stronger than Theorem 1.

Theorem 1 can be restated as follows: every graph with is -choosable with defect , whereas Theorem 2 says that is -choosable with defect . Both results require that , and the minimum value of for which either theorem is applicable is . In this case, Theorem 2 gives a defect bound of , which is an order of magnitude less than the defect bound of in Theorem 1. Note that Havet and Sereni [27] gave a construction to show that no lower value of is possible. That is, for , the defective chromatic number of the class of graphs with maximum average degree equals ; also see [42].

See [33, 10, 11, 32, 8, 7, 6, 5, 4, 9] for results about defective 2-colourings of graphs with given maximum average degree, where each of the two colour classes has a prescribed degree bound. Also note that Dorbec et al. [17] proved a result analogous to Theorems 2 and 1 (with weaker bounds) for defective colouring of graphs with given maximum average degree, where in addition, a given number of colour classes are stable sets.

1.2. Clustered Choosability

The following theorem, due to Kopreski and Yu [34], is the only known non-trivial result for clustered colourings of graphs with given maximum average degree111Kopreski and Yu [34] actually proved the following stronger result: For and , every graph with is -colourable, such that colour classes have defect , and colour classes are stable sets..

Theorem 3 ([34]).

Every graph is -colourable with defect , and thus with clustering .

There are no existing non-trivial results for clustered choosability of graphs with given maximum average degree. The closest such result, due to Dvořák and Norin [21], says that for constants , if a graph has at most edges, and every -vertex subgraph of has a balanced separator of order at most , then is -choosable with clustering some function of , and . Note that the number of colours here is roughly half the average degree of . This result determines the clustered chromatic number of several graph classes, but for various other classes (that contain expanders) this result is not applicable because of the requirement that every subgraph has a balanced separator.

Theorem 2 with implies the above result of Kopreski and Yu [34] and extends it to the setting of choosability:

Theorem 4.

Every graph is -choosable with defect , and thus with clustering .

As an example of Theorem 4, it follows from Euler’s formula that toroidal graphs have maximum average degree at most 6, implying every toroidal graph is 5-choosable with defect and clustering , which was first proved by Dujmović and Outioua [18]. Previously, Cowen et al. [14] proved that every toroidal graph is 5-colourable with defect .

The following two theorems are our main results for clustered choosability. The first still has an absolute bound on the clustering, while the second has fewer colours but at the expense of allowing the clustering to depend on the maximum average degree.

Theorem 56).

Every graph is -choosable with clustering .

Theorem 67).

Every graph is -choosable with clustering .

Theorem 6 says that the clustered chromatic number of the class of graphs with maximum average degree is at most . This is the best known upper bound. The best known lower bound is ; see [42]. Closing this gap is an intriguing open problem.

1.3. Generalisation

The above results generalise via the following definition. For a graph and integer , let be the maximum average degree of a subgraph of with at least vertices, unless , in which case . The next two results generalise Theorems 6 and 2 respectively with replaced by , where the number of colours stays the same, and the defect or clustering bound also depends on .

Theorem 73).

For integers , and , every graph with is -choosable with defect .

Theorem 87).

For integers , and , every graph with is -choosable with clustering .

Note that Theorem 7 with is equivalent to Theorem 2, and Theorem 8 with and is equivalent to Theorem 6.

Graphs on surfaces provide motivation for this extension222The Euler genus of the orientable surface with handles is . The Euler genus of the non-orientable surface with cross-caps is . The Euler genus of a graph is the minimum Euler genus of a surface in which embeds.. Graphs with Euler genus can have average degree as high as , the complete graph being one example. But such graphs necessarily have bounded size. In particular, Euler’s formula implies that every -vertex -edge graph with Euler genus satisfies . Thus, for , if then has average degree . In particular, .

Using this observation, Theorems 8 and 7 respectively imply that graphs with bounded Euler genus are -choosable with bounded defect and are -choosable with bounded clustering. Both these results are actually weaker than known results. In particular, several authors [3, 14, 43, 13] have proved that graphs with bounded Euler genus are 3-colourable or 3-choosable with bounded defect. And Dvořák and Norin [21] proved that graphs with bounded Euler genus are 4-choosable with bounded clustering. The proof of Dvořák and Norin [21] uses the fact that graphs of bounded Euler genus have strongly sub-linear separators. The advantage of our approach is that it works for graph classes that do not have sub-linear separator theorems. Graphs with given -thickness are such a class [19]. We explore this direction in Section 8.

1.4. Clustered Choosability and Maximum Degree

Alon, Ding, Oporowski, and Vertigan [2] and Haxell, Szabó, and Tardos [28] studied clustered colourings of graphs with given maximum degree. Haxell et al. [28] proved that every graph with maximum degree is -colourable with bounded clustering. Moreover, for some and , every graph with maximum degree is -colourable with bounded clustering. For both these results, the clustering bound is independent of .

Clustered choosability of graphs with given maximum degree has not been studied in the literature (as far as we are aware). As a by-product of our work for graphs with given maximum average degree we prove the following results for clustered choosability of graphs with given maximum degree.

Theorem 95).

Every graph with maximum degree is -choosable with clustering .

Theorem 106).

Every graph with maximum degree is -choosable with clustering .

is the first case in which the above results for clustered choosability are weaker than the known results for clustered colouring. In particular, Haxell et al. [28] proved that every graph with maximum degree is -colourable with bounded clustering, whereas Theorems 10 and 9 only prove 3-choosability. It is open whether every graph with maximum degree is -choosable with bounded clustering.

Finally, we remark that all our choosability results hold in the stronger setting of correspondence colouring, introduced by Dvořák and Postle [22].

2. Definitions

Let be a graph with vertex set and edge set . Let be the maximum degree of the vertices in . For a subset and vertex , let and . We sometimes refer to as .

In a coloured graph, the defect of a vertex is its degree in the monochromatic subgraph. Note that a colouring with defect also has defect , but a vertex of defect does not have defect .

3. Defective Choosability and Maximum Average Degree

This section proves our result for defective choosability (Theorem 2). The following lemma is essentially a special case of an early result of Lovász [36].

Lemma 11.

If is a list-assignment for a graph , such that

for each vertex of , then is -colourable with defect .

Proof.

Colour each vertex in by a colour in so that the number of monochromatic edges is minimised. Suppose that some vertex coloured is adjacent to at least vertices also coloured . Since , some colour is assigned to at most neighbours of . Recolouring by reduces the number of monochromatic edges. This contradiction shows that no vertex is adjacent to at least vertices of the same colour as . Thus the colouring has defect . ∎

Corollary 12.

Every graph with is -choosable with defect .

The next lemma is a key idea of this paper. It provides a sufficient condition for a partial list-colouring to be extended to a list-colouring of the whole graph.

Lemma 13.

Let be a -list-assignment of a graph . Let be a partition of , where is -colourable with defect . If and for every vertex ,

then is -colourable with defect .

Proof.

Let be an -colouring of with defect . For each vertex , let . Thus . Lemma 11 implies that is -colourable with defect . By construction, there is no monochromatic edge between and . Thus is -colourable with defect . ∎

We now prove our first main result, which is equivalent to Theorem 2 when .

See 7

Proof.

We proceed by induction on . Let be a -list-assignment for . For the base case, suppose that . For each vertex of , choose a colour in so that each colour is used at most times. We obtain an -colouring with defect . Now assume that .

Let be a maximal sequence of distinct vertices in , such that for all , we have , where and .

First suppose that . Let and . By induction, is -colourable with defect . By the maximality of , for every vertex , we have . By Lemma 13, is -colourable with defect , and we are done.

Now assume that . Thus

Since , we have , which is a contradiction. ∎

4. Using Independent Transversals

This section introduces a useful tool, called “independent transversals”, which have been previously used for clustered colouring by Alon et al. [2] and Haxell et al. [28]. Haxell [29] proved the following result.

Lemma 14 ([29]).

Let be a graph with maximum degree at most . Let be a partition of , with for each . Then has a stable set with for each .

Lemma 15.

Let and let be a graph of maximum degree at most . If is a subgraph of with , then has a stable set of vertices of degree 2 in with the following properties:

  1. every subpath of with at least vertices that contains a vertex with degree 1 in contains at least one vertex in ,

  2. every subpath of with at least vertices that contains a vertex with degree 1 in contains at least two vertices in ,

  3. every connected subgraph of with at least vertices contains at least three vertices in .

Proof.

Consider each cycle component of with . Say , where and . Partition into subpaths where and for . Note that .

Now consider each path component of with . Say , where and . Partition into subpaths where for , for , and .

Let be the set of all such paths taken over all the components of . Let . Then gives a partition of into parts, each of which has exactly vertices, and . By Lemma 14, has a stable set that contains exactly one vertex in each path in . By construction, every vertex in has degree 2 in and is a stable set in , so is a stable set in .

Let be a path in that contains a vertex of degree 1 in . Then is subpath of some component path of . If contains at least vertices, then where and . Now, using our previous notation, and , so is not a proper subpath of or of . Hence contains every vertex of for some , so contains a vertex in .

If contains at least vertices, then where and . Now, and , so is not a proper subpath of or of . Hence contains every vertex and of for some , so contains two vertices in .

Suppose for contradiction there is a connected subgraph of on vertices with at most two vertices in . By the definition of , there are at most two paths with . If is contained in some path component of , then is a proper subpath of for some , where we take and to be the empty path for simplicity (so and ). Now .

If is contained in some cycle component of , we may assume without loss of generality that is a subpath of the path , and does not contain every vertex of and does not contain every vertex of . Thus, , a contradiction. ∎

5. Clustered Choosability and Maximum Degree

This section proves our first result about clustered choosability of graphs with given maximum degree (Theorem 9). The preliminary lemmas will also be used in subsequent sections.

Lemma 16.

If is a list-assignment for a graph , such that for each vertex of , and is an -colouring of that minimises the number of monochromatic edges, then has defect 2. Moreover, for each vertex with defect 2 under , there is a colour , such that at most two neighbours of are coloured under .

Proof.

Suppose that some vertex coloured is adjacent to at least three vertices also coloured . Since , some colour is assigned to at most two neighbours of . Recolouring by reduces the number of monochromatic edges. This contradiction shows that every vertex has defect at most .

Consider a vertex coloured with defect 2. Suppose that has at least three neighbours coloured for each . Thus , implying , which is a contradiction. Thus some colour is assigned to at most two neighbours of . ∎

Given a colouring of a graph , let denote the monochromatic subgraph of given . The idea for the following lemma is by Haxell et al. [28, Lemma 2.6], adapted here for the setting of list-colourings.

Lemma 17.

If is a bipartite graph with bipartition and is a list-assignment for such that for all and for all and every -colouring has defect 2, then has an -colouring such that every connected subgraph of at most two vertices in .

Proof.

We begin by orienting the edges of so that for every vertex and every colour , has at most one out-neighbour with and has at most one in-neighbour with . Let be the union of the lists of all vertices of . For each colour , let be the subgraph of induced by the vertices with . There is an -colouring which assigns each vertex of the colour , so . Also, since every edge of has an endpoint and , evey edge of is in for at most one . For each , orient the edges of so that no vertex has more than one in-neighbour or out-neighbour (possible since ). Orient all remaining edges of arbitrarily.

We now construct an -colouring . First, colour each vertex in with the unique colour in its list. Now run the following procedure, initialising .

  1. If , then exit.

  2. Select and select arbitrarily. Increment by and go to 3.

  3. If there is a directed path such that and and , let , select , increment by and go to 3. Otherwise go to 4.

  4. If there is a directed path such that and and , let , select , increment by and go to 3. Otherwise go to 1.

Suppose for contradiction that some component of has at least three vertices in . Since is an -colouring, has a directed subpath such that . If was the first vertex in to be coloured, then was coloured next and , a contradiction. If was the first vertex in to be coloured, then was coloured next and , a contradiction. Hence, was coloured before and after . But then was coloured immediately after and , a contradiction. ∎

We now prove our first result for clustered choosability of graphs with given maximum degree.

See 9

Proof.

Let . Let be a -list-assignment for . Let be an -colouring of that minimises the number of monochromatic edges. By Lemma 16, is an -colouring with defect 2. Moreover, for each vertex with defect 2 under , there is a colour , such that at most two neighbours of are coloured under . Let for each vertex with defect 2.

Let be the monochromatic subgraph of . Thus . By Lemma 15, there is a set , such that is stable in , every vertex in has defect 2 under , and the following hold:

  1. every subpath of with at least vertices that contains a vertex with degree 1 in contains at least one vertex in ,

  2. every subpath of with at least vertices that contains a vertex with degree 1 in contains at least two vertices in , and

  3. every connected subgraph of on at least vertices contains at least three vertices in .

Define a subpath of to have type 1 if it contains no vertex in and at least one vertex of degree at most 1 in . Define a subpath of to have type 2 if it contains at most one vertex in and at least one vertex of degree at most 1 in . Note that every path of type 1 is also of type 2, and every path of type 2 or 1 that does not contain a vertex of degree 1 in contains a vertex of degree 0 in , and hence has only one vertex. By the definition of , every path of type 1 has at most vertices and every path of type 2 has at most vertices.

Let be the set of connected components of . Let be the bipartite graph with bipartition , where is adjacent to if and only if is adjacent to in , and the colour of the vertices of is in . Define so that for every , and is the singleton containing the colour assigned to the vertices of for every .

Let be an arbitrary -colouring of , and let be the corresponding -colouring of . Note that every vertex of is assigned a colour in and every other vertex is assigned its original colour in . Since is a stable set and by the definition of , the number of monochromatic edges given is at most the number of monochromatic edges given . Hence by our choice of , no -colouring of yields fewer monochromatic edges than . Hence the monochromatic subgraph of given satisfies . Let be the graph obtained from by contracting each to a single vertex. Then is isomorphic to the monochromatic subgraph of given . Since is a minor of and , we have . Hence, every -colouring of has defect 2.

By Lemma 17, has an -colouring such that no component of the monochromatic subgraph has more than two vertices in . Let be the corresponding -colouring of , and note that no component of the monochromatic subgraph of given has more than two vertices in . In , vertices of keep their colour from , and vertices get a colour from , so is an -colouring that minimises the number of monochromatic edges.

Suppose for contradiction that some vertex in has degree 2 in and is adjacent in to some vertex which is not its neighbour in (so ). Then the -colouring obtained from by recolouring with is not 2-defective, a contradiction.

It follows that the largest possible monochromatic component of is obtained either from three disjoint paths in of type 1 linked by two vertices in , or is obtained from a path of type 1 and a path of type 2 linked by a vertex of , or is a subgraph of that contains at most two vertices in . In each case, we have . ∎

6. Clustered Choosability with Absolute Bounded Clustering

This section proves our results for clustered choosability of graphs with given maximum average degree (Theorem 5) or given maximum degree (Theorem 10), where the clustering is bounded by an absolute constant. The following lemma is the heart of the proof. With , it immediately implies Theorem 10.

Lemma 18.

If is a stable set of vertices in a graph and is a list-assignment for such that for all and for all , then has an -colouring with clustering 9. Furthermore, if , then has an -colouring with clustering .

Proof.

Let be the class of -colourings that minimise the number of monochromatic edges. Given and , let be the set of colours such the colouring obtained from by recolouring with is in . Note that in particular , and that a colour is in if and only if .

Claim 1.

If , then .

Proof.

Let be a vertex of maximum degree in . If for some colour we have , then the colouring obtained from by changing the colour of to satisfies , contradicting the assumption that . Hence, . By assumption , and the result follows. ∎

Claim 2.

If , and , then .

Proof.

Suppose for contradiction that . Note that . Given that , we have without loss of generality. Since , for every colour , the vertex has at least two neighbours in coloured by (or else recolouring with would yield a colouring with ). For every colour , the vertex has at least three neighbours coloured by . Hence, , meaning , a contradiction. ∎

Choose and such that is a stable set in , every vertex in has degree 2 in , and subject to this is maximised. Let . For , define recursively so that for and . Such -colourings exist by Claim 2.

Define for all .

Claim 3.

If is an -colouring of and , then .

Proof.

Note that for . Hence if . Now suppose that . By construction, , so the colouring obtained from by changing the colour of to is in . Now by Claim 1, so no vertex is adjacent to in , since already has two neighbours in and hence in . Since , we have . Hence . ∎

Claim 4.

If is an -colouring of , then .

Proof.

Suppose for contradiction that for some , . Since is a stable set in , either or .

If and , then has three neighbours in by Claim 3. But since for , we have , a contradiction.

Hence, without loss of generality, and . Now , so the colouring obtained from by recolouring with is in . Note by assumption. By Claim 3, , so , contradicting Claim 1.

Now by Claim 3. But , so , and . ∎

Let be the set of components of . Let be the bipartite graph with bipartition such that is adjacent to if is adjacent to in and the colour assigned to the vertices of by is in . Let be the natural restriction of to . Note that an -colouring of corresponds to an -colouring of , and is a minor of , which means by Claims 4 and 1. Hence, by Lemma 17, has an -colouring such that no component of has more than two vertices in . Let be the corresponding -colouring of . Note that each component of has at most two vertices in .

Suppose for contradiction that some component of has at least ten vertices. Now by Claims 4 and 1, so is a cycle or a path. Hence has an induced subpath such that every vertex of has degree 2 in . Since is a stable set in , at most one vertex in each of , and is in , so