Clustered Colouring in Minor-Closed Classes
Abstract. The clustered chromatic number of a class of graphs is the minimum integer such that for some integer every graph in the class is -colourable with monochromatic components of size at most . We prove that for every graph , the clustered chromatic number of the class of -minor-free graphs is tied to the tree-depth of . In particular, if is connected with tree-depth then every -minor-free graph is -colourable with monochromatic components of size at most . This provides the first evidence for a conjecture of Ossona de Mendez, Oum and Wood (2016) about defective colouring of -minor-free graphs. If then we prove that 4 colours suffice, which is best possible. We also determine those minor-closed graph classes with clustered chromatic number 2. Finally, we develop a conjecture for the clustered chromatic number of an arbitrary minor-closed class.
In a vertex-coloured graph, a monochromatic component is a connected component of the subgraph induced by all the vertices of one colour. A graph is -colourable with clustering if each vertex can be assigned one of colours such that each monochromatic component has at most vertices. We shall consider such colourings, where the first priority is to minimise the number of colours, with small clustering as a secondary goal. With this viewpoint the following definition arises. The clustered chromatic number of a graph class , denoted by , is the minimum integer such that, for some integer , every graph in has a -colouring with clustering . See  for a survey on clustered graph colouring.
This paper studies clustered colouring in minor-closed classes of graphs. A graph is a minor of a graph if a graph isomorphic to can be obtained from some subgraph of by contracting edges. A class of graphs is minor-closed if for every graph every minor of is in , and some graph is not in . For a graph , let be the class of -minor-free graphs (that is, not containing as a minor). Note that we only consider simple finite graphs.
As a starting point, consider Hadwiger’s Conjecture, which states that every graph containing no -minor is properly -colourable. This conjecture is easy for , is equivalent to the 4-colour theorem for , is true for , and is open for . The best known upper bound on the chromatic number is , independently due to Kostochka [9, 10] and Thomason [20, 21]. This conjecture is widely considered to be one of the most important open problems in graph theory; see  for a survey.
Clustered colourings of -minor-free graphs provide an avenue for attacking Hadwiger’s Conjecture. Kawarabayashi and Mohar  first proved a upper bound on . In particular, they proved that every -minor-free graph is -colourable with clustering , for some function . The number of colours in this result was improved to by Wood , to by Edwards, Kang, Kim, Oum, and Seymour , to by Liu and Oum , and to by Norin . Thus . See [7, 6] for analogous results for graphs excluding odd minors. For all of these results, the function is very large, often depending on constants from the Graph Minor Structure Theorem. Van den Heuvel and Wood  proved the first such result with explicit. In particular, they proved that every -minor-free graph is -colourable with clustering . The result of Edwards et al.  mentioned below implies that . Dvořák and Norin  have announced a proof that .
Now consider the class of -minor-free graphs for an arbitrary graph . The maximum chromatic number of a graph in is at most and is at least (since is -minor-free), and Hadwiger’s Conjecture would imply that is the answer. However, for clustered colourings, fewer colours often suffice. For example, Dvořák and Norin  proved that graphs embeddable on any fixed surface are 4-colourable with bounded clustering, whereas the chromatic number is for surfaces of Euler genus . Van den Heuvel and Wood  proved that -minor-free graphs are 3-colourable with clustering , and that -minor-free graphs are 6-colourable with clustering . These results show that depends on the structure of , unlike the usual chromatic number which only depends on .
At the heart of this paper is the following question: what property of determines ? The following definitions help to answer this question. Let be a rooted tree. The depth of is the maximum number of vertices on a root–to–leaf path in . The closure of is obtained from by adding an edge between every ancestor and descendent in . The connected tree-depth of a graph , denoted by , is the minimum depth of a rooted tree such that is a subgraph of the closure of . This definition is a variant of the more commonly used definition of the tree-depth of , denoted by , which equals the maximum connected tree-depth of the connected components of . See  for background on tree-depth. If is connected, then . In fact, unless has two connected components and with , in which case . We choose to work with connected tree-depth to avoid this distinction.
The following result is the primary contribution of this paper; it is proved in Section 2.
For every graph , is tied to the (connected) tree-depth of . In particular,
The upper bound in Theorem 1 gives evidence for, and was inspired by, a conjecture of Ossona de Mendez, Oum, and Wood , which we now introduce. A graph is -colourable with defect if each vertex of can be assigned one of colours so that each vertex is adjacent to at most neighbours of the same colour; that is, each monochromatic component has maximum degree at most . The defective chromatic number of a graph class , denoted by , is the minimum integer such that, for some integer , every graph in is -colourable with defect . Every colouring of a graph with clustering has defect . Thus the defective chromatic number of a graph class is at most its clustered chromatic number. Ossona de Mendez et al.  conjectured the following behaviour for the defective chromatic number of .
Conjecture 2 ().
For every graph ,
Ossona de Mendez et al.  proved the lower bound, , in Conjecture 2. This follows from the observation that the closure of the rooted complete -ary tree of depth is not -colourable with clustering . The lower bound in Theorem 1 follows since for every class. The upper bound in Conjecture 2 is known to hold in some special cases. Edwards et al.  proved it if ; that is, , which can be thought of as a defective version of Hadwiger’s Conjecture. Ossona de Mendez et al.  proved the upper bound in Conjecture 2 if or if is a complete bipartite graph. In particular, .
While it is conjectured that , the following lower bound, proved in Section 2.3, shows that might be larger, thus providing some distinction between defective and clustered colourings.
For each , there is a graph with such that
We conjecture an analogous upper bound:
For every graph ,
A further contribution of the paper is to precisely determine the minor-closed graph classes with clustered chromatic number 2. This result is introduced and proved in Section 3. Section 4 studies clustered colourings of graph classes excluding so-called fat stars as a minor. This leads to a proof of Conjecture 4 in the case. We conclude in Section 5 with a conjecture about the clustered chromatic number of an arbitrary minor-closed class that generalises Conjecture 4.
2 Tree-depth Bounds
The main goal of this section is to prove that is bounded from above by some function of . We actually provide two proofs. The first proof depends on deep results from graph structure theory and gives no explicit bound on the clustering. The second proof is self-contained, but gives a worse upper bound on the number of colours. Both proofs have their own merits, so we include both.
Let be the closure of the rooted complete -ary tree of depth . (Here each non-leaf node has exactly children.)
If is a vertex in a connected graph and for , then is called the BFS layering of starting at .
2.1 First Proof
The first proof depends on the following Erdős-Pósa Theorem by Robertson and Seymour . For a graph and integer , let be the disjoint union of copies of .
For every non-empty graph with connected components and for all integers , for every graph with treewidth at most and containing no minor, there is a set of size at most such that has no minor.
The next lemma is the heart of our proof.
For all integers , every -minor-free graph of treewidth at most is -colourable with clustering .
We proceed by induction on , with and fixed. The case is trivial since is the 1-vertex graph, so only the empty graph has no minor, and the empty graph is 0-colourable with clustering 0. Now assume that , the claim holds for , and is a -minor-free graph with treewidth at most . Let be the BFS layering of starting at some vertex .
Fix . Then contains no as a minor, as otherwise contracting to a single vertex gives a minor (since every vertex in has a neighbour in ). Since has treewidth at most , so does . By Theorem 5 with and , there is a set of size at most , such that has no minor. By induction, is -colourable with clustering . Use one new colour for . Thus is -colourable with clustering .
Use disjoint sets of colours for even and odd , and colour by one of the colours used for even . No edge joins with for . Thus is -coloured with clustering . ∎
To drop the assumption of bounded treewidth, we use the following result of DeVos, Ding, Oporowski, Sanders, Reed, Seymour, and Vertigan , the proof of which depends on the graph minor structure theorem.
Theorem 7 ().
For every graph there is an integer such that for every graph containing no -minor, there is a partition of such that has treewidth at most , for .
For all integers , there is an integer , such that every -minor-free graph is -colourable with clustering at most .
Fix a graph . By definition, is a subgraph of . Thus every -minor-free graph contains no -minor. Hence, Lemma 8 implies
which is the upper bound in Theorem 1.
2.2 Second Proof
Let be a rooted tree. Recall that the closure of is the graph with vertex set , where two vertices are adjacent in if one is an ancestor of the other in . The weak closure of is the graph with vertex set , where two vertices are adjacent in if one is a leaf and the other is one of its ancestors. For , let be the rooted complete -ary tree of depth . Let be the weak closure of .
For , the graph contains as a minor.
Let be the root vertex. Colour blue. For each non-leaf vertex , colour children of blue and colour the other child of red. Let be the set of blue vertices in , such that every ancestor of is blue. Note that induces a copy of in . Let be a non-leaf vertex in . Let be the red child of , and let be the subtree of rooted at . Then every leaf of is adjacent in to and to every ancestor of . Contract and the edge into . Now is adjacent to every ancestor of in . Do this for each non-leaf vertex in . Note that and are disjoint for distinct non-leaf vertices . Thus, we obtain as a minor of . ∎
A model of a graph in a graph is a collection of pairwise disjoint subtrees of such that for every there is an edge of with one end in and the other end in . Observe that a graph contains as a minor if and only if it contains a model of .
For and , if a graph contains as a minor, then contains subgraphs and , both containing as a minor, such that .
Let be a model of in . Let be the root vertex of . We may assume that for each leaf vertex of , there is exactly one edge between and .
Let be a tree obtained from by splitting vertices, where:
has maximum degree at most 3,
is a minor of ; let be the model of in , so each edge of corresponds to an edge of between and ,
there is a set of leaf vertices in , and a bijection from to the set of leaves of , such that for each leaf of , if the edge between and in is incident with vertex in , then is a vertex in , in which case we say and are associated.
Let . Apply the following ‘propagation’ process in . Initially, say that the vertices in are alive with respect to . For each parent vertex of leaves in , if at least of its children are alive with respect to , then is also alive with respect to . Now propagate up , so that a non-leaf vertex of is alive if and only if at least of its children are alive with respect to . Say is good if is alive with respect to .
For an edge of let be the set of vertices in in the subtree of containing , and let be the set of vertices in in the subtree of containing . Since is the disjoint union of and , every leaf vertex of is in exactly one of or . By induction, every vertex in is alive with respect to or (possibly both). In particular, or is good (possibly both).
Suppose that both and are good. Then at least children of are alive with respect to , and at least children of are alive with respect to . Thus there are disjoint sets and , each consisting of children of , where every vertex in is alive with respect to , and every vertex in is alive with respect to . We now define a set of vertices, said to be chosen by , all of which are alive with respect to . First, each vertex in is chosen by . Then for each non-leaf vertex chosen by , choose children of that are also alive with respect to , and say they are chosen by . Continue this process down to the leaves of . We now define the graph , which is initially empty. For each vertex chosen by , add the subgraph to . Furthermore, for each leaf vertex of chosen by and for each ancestor of chosen by , add the edge in between and to . Define analogously with respect to and . At this point, and are disjoint.
The edge in either corresponds to an edge or a vertex of . First suppose that corresponds to an edge of , where is in and is in . Let be the subtree of containing . Add to , plus the edge in between and for each leaf of chosen by . Similarly, let be the subtree of containing , and add to , plus the edge in between and for each leaf of chosen by . Observe that and are disjoint, and they both contain as a minor, as desired.
Now consider the case in which corresponds to a vertex in ; that is, and are both in . Let be the subtree of corresponding to the subtree of containing (which includes ). Add to , plus the edge in between and for each leaf of chosen by . Similarly, let be the subtree of corresponding to the subtree of containing (which includes ). Add to , plus the edge in between and for each leaf of chosen by . Observe that both and contain as a minor, and , as desired.
We may therefore assume that for each edge of , exactly one of and is good. Orient towards if is good, and towards if is good. Since at most one leaf of is associated with each leaf of , each edge incident with a leaf of is oriented away from the leaf. Since is a tree, contains a sink vertex , which is therefore not a leaf. Let , and possibly be the neighbours of in . Let be the set of vertices in in the subtree of containing . Since is oriented towards , with respect to , the set is not good. Since no leaf of is associated with , the sets , and partition the leaves of . Since each non-leaf vertex in has children, is alive with respect to at least one of , or . In particular, at least one of , or is good. This is a contradiction. ∎
Let for every . Then there is a function such that for every , every graph either contains as a minor or is -colourable with clustering .
We proceed by induction on . In the base case, , since is the 1-vertex graph, the result holds with . Now assume that and the result holds for and all .
Let be a graph, which we may assume is connected. Let be a BFS layering of .
Fix . Let be the maximum integer such that contains disjoint subgraphs each containing a minor. First suppose that . Then contains disjoint subgraphs each containing a minor. Contracting to a single vertex gives a minor (since every vertex in has a neighbour in ), and we are done. Now assume that .
If , then contains no minor. By induction, is -colourable with clustering .
Now consider the case that . Apply Lemma 10 to for each . Thus contains subgraphs and , both containing as a minor, such that . Let . Thus . Let and . By the maximality of , the subgraph contains no minor (as otherwise would give pairwise disjoint subgraphs satisfying the requirements). By induction, is -colourable with clustering since . Similarly, is -colourable with clustering . By construction, each vertex in is in at least one of , or . Use one new colour for , which has size at most .
In both cases, is -colourable with clustering . Use a different set of colours for even and for odd , and colour by one of the colours used for even . No edge joins with for . Since , is -colourable with clustering . ∎
Note that the clustering function in Theorem 11 satisfies
where the number of s is .
For every graph ,
2.3 Lower Bound
We now prove Theorem 3, where , the closure of the complete ternary tree of depth (which has tree-depth and connected tree-depth ).
Fix an integer . We now recursively define graphs (depending on ), and show by induction on that has no -colouring with clustering , and is not a minor of .
For the base case , let be the path on vertices. Then has no minor, and has no 1-colouring with clustering .
Assume is defined for some , that has no -colouring with clustering , and is not a minor of . As illustrated in Figure 1, let be obtained from a path as follows: for add pairwise disjoint copies of complete to .
Suppose that has a -colouring with clustering . Then and receive distinct colours for some . Consider the copies of complete to . At most such copies contain a vertex assigned the same colour as , and at most such copies contain a vertex assigned the same colour as . Thus some copy avoids both colours. Hence is -coloured with clustering , which is a contradiction. Therefore has no -colouring with clustering .
It remains to show that is not a minor of . Suppose that contains a model of . Let be the root vertex in . Choose the -model to minimise . Since induces a connected dominating subgraph in , by the minimality of the model, is a connected subgraph of . Say . Note that consists of three pairwise disjoint copies of . The model of one such copy avoids and (if these vertices are defined). Since is connected, is contained in a component of and is adjacent to . Each such component is a copy of . Thus is a minor of , which is a contradiction. Thus is not a minor of . ∎
3 2-Colouring with Bounded Clustering
This section considers the following question: which minor-closed graph classes have clustered chromatic number 2? To answer this question we introduce three classes of graphs that are not 2-colourable with bounded clustering, as illustrated in Figure 2.
The first example is the -fan, which is the graph obtained from the -vertex path by adding one dominant vertex. If the -fan is 2-colourable with clustering , then the underlying path contains at most vertices of the same colour as the dominant vertex, implying that the other colour has at most monochromatic components each with at most vertices, and . That is, if then the -fan is not 2-colourable with clustering .
The second example is the -fat star, which is the graph obtained from the -star (the star with leaves) as follows: for each edge in the -star, add degree-2 vertices adjacent to and . Note that the -fat star is . Suppose that the -fat star has a 2-colouring with clustering . Deleting the dominant vertex in the -fat star gives disjoint -stars. Since , in at least one of these -stars, no vertex receives the same colour as the dominant vertex, implying there is a monochromatic component on vertices. Thus, for there is no 2-colouring of the -fat star with clustering .
The third example is the -fat path, which is the graph obtained from the -vertex path as follows: for each edge of the -vertex path, add degree-2 vertices adjacent to and . If then in every 2-colouring of the -fat path with clustering , adjacent vertices in the underlying path receive the same colour, implying that the underlying path is contained in a monochromatic component with more than vertices. Thus, for there is no 2-colouring of the -fat path with clustering .
These three examples all need three colours in a colouring with bounded clustering. The main result of this section is the following converse result.
Let be a minor-closed graph class. Then if and only if for some integer , the -fan, the -fat path, and the -fat star are not in .
Lemma 24 below shows that every graph containing no -fan minor, no -fat path minor, and no -fat star minor is 2-colourable with clustering for some explicit function . Along with the above discussion, this implies Theorem 14. We assume for the remainder of this section.
The following definition is a key to the proof. For an -vertex graph with vertex set , a -strong -model in a graph consists of pairwise disjoint connected subgraphs in , such that for each edge of there are at least vertices in adjacent to both and . Note that a vertex in might count towards this set of vertices for distinct edges of . This definition leads to the following sufficient condition for a graph to contain a -fat star or -fat path
If a graph contains a -strong -model for some connected graph with edges, then contains a -fat star or a -fat path as a minor.
Use the notation introduced in the definition of -strong -model. Since is connected with edges, contains a -vertex path or a -leaf star as a subgraph. Suppose that is a -vertex path in . For , let be a set of vertices in
each of which is adjacent to both and . Such a set exists since and have at least common neighbours in . For , contract one vertex of into . Then contract each of into a single vertex. We obtain the -fat path as a minor in . The case of a -leaf star is analogous. ∎
If a connected graph contains a -strong -model, for some graph with connected components, then contains a -strong -model for some connected graph with .
We proceed by induction on . The case is vacuous. Assume , and the result holds for . Let be the components of . We may assume that has no isolated vertices. Say is a -strong -model in . For each edge in , let be a set of common neighbours of and . For each component of , note that induces a connected subgraph in , which we denote by . Since is connected, there is a path between and , for some distinct , such that no internal vertex of is in . Note that might be a single vertex. For some edge in and some edge in , without loss of generality, joins some vertex in and some vertex in . Let be the graph obtained from by identifying and into a new vertex . Now has components and . Define . If then add the edge between and to . Similarly, if then add the edge between and to . Remove and/or from for each edge of . Now . We obtain a -strong -model in . By induction, contains a -strong -model for some connected graph with . ∎
If a connected graph contains a -strong -model for some graph with at least edges, then contains a -fat star or a -fat path as a minor.
Let be a connected graph such that for some non-cut-vertex and integers . Then contains a -fan as a minor, or contains a connected subgraph and has neighbours not in and all adjacent to (thus contracting gives a minor).
Let be a vertex of . For each , let be a -path in . If for some , then contains a -fan minor. Now assume that for each . Let be the digraph with vertex set , where for each vertex . Thus has maximum outdegree at most , and the underlying undirected graph of has average degree at most . Since , by Turán’s Theorem, contains a stable set of size . Let , which is connected since is stable. Each vertex in is adjacent to and to , as desired. ∎
Let be a graph with distinct vertices