Islands in minor-closed classes. I. Bounded treewidth and separators
The clustered chromatic number of a graph class is the minimum integer such that for some the vertices of every graph in the class can be colored in colors so that every monochromatic component has size at most . We show that the clustered chromatic number of the class of graphs embeddable on a given surface is four, proving the conjecture of Esperet and Ochem. Additionally, we study the list version of the concept and characterize the minor-closed classes of graphs of bounded treewidth with given clustered list chromatic number. We further strengthen the above results to solve some extremal problems on bootstrap percolation of minor-closed classes.
Let and be graphs. A model of in is a function assigning to vertices of pairwise vertex-disjoint non-empty connected subgraphs of such that for every , there exists an edge of with one end in and the other end in . If has a model in , we say that is a minor of , otherwise, we say that is -minor-free. A class of graphs is a (proper) minor-closed class if does not contain all graphs, and for every graph every minor of also belongs to . We define the chromatic number of a graph class as the minimum integer such that every graph is properly -colorable (and if no such exists). The famous Hadwiger’s conjecture can be considered as a characterization of minor-closed graph classes with given chromatic number.
Conjecture 1 (Hadwiger’s conjecture ).
Let be an integer, and let be a minor-closed class of graphs. Then if and only if .
For integers and , a -coloring with clustering of a graph is a (not necessarily proper) coloring of vertices of using colors
such that contains no monochromatic connected subgraph with more than vertices. We denote by the minimum such that admits a -coloring with clustering . The clustered chromatic number of a graph class is the minimum such that there exists so that for every graph
( if no such exists).
The clustered chromatic number of minor-closed classes of graphs has been recently actively investigated, motivated in part by Hadwiger’s conjecture.
Let denote the minor-closed class consisting of all -minor-free graphs. Kawarabayashi and Mohar  proved the first linear bound on showing that .
The above definitions can be straightforwardly extended from coloring to list coloring, and many of our techniques extend as well. A -list assignment for a graph assigns a finite set of size at least to every vertex . An -coloring with clustering of a graph assigns to every vertex a color from such that contains no monochromatic connected subgraph with more than vertices. Let denote the minimum such that admits an -coloring with clustering for every -list assignment . Clearly, . The clustered list chromatic number of a graph class is the minimum such that there exists so that for every graph (and if no such exists).
In the sequel to this paper
we show that , proving the weakening of Hadwiger’s conjecture for clustered chromatic number.
Rather than working with the clustered (list) chromatic number, we bound a different parameter which dominates it. The coloring number of a graph is the smallest integer such that every non-empty subgraph of contains a vertex of degree less than . A standard greedy argument shows that . Let us now introduce a similar notion for clustered coloring, first defined by Esperet and Ochem . A -island in a graph is a non-empty subset of vertices of such that each vertex of has less than neighbors in . Let denote the smallest integer such that each non-empty subgraph of contains a -island of size at most . Hence, . The following analogue of the bound holds for the coloring with bounded clustering.
For every integer , every graph satisfies .
Let be a -list assignment for . Suppose that each non-empty subgraph of contains a -island of size at most . By induction on the size of , we show that has an -coloring with clustering at most . This is trivial if is empty, hence assume that . Let be a -island in of size at most . By the induction hypothesis, has an -coloring with component size at most . Color each vertex of by an arbitrary color from different from the colors of its neighbors in . Clearly, each connected monochromatic subgraph of is contained either in or in , and since , we conclude that we obtained an -coloring of with clustering at most . ∎
For a class of graphs, let the clustered coloring number of denote the smallest integer such that there exists such that for every graph ( if no such exists). By Lemma 2, , i.e. any upper bound on gives an upper bound on . This observation motivates our investigation of the clustered coloring number in this paper.
It follows from Observation 3 below that . On the other hand, for every , there exist minor-closed graph classes containing with , e.g., the class of all graphs with at most vertices. Thus the complete minors present in the minor-closed class do not determine the clustered coloring number of the class. This motivates us to ask for an exact description of minor-closed classes with .
Two kinds of graphs seem to play an important role in this context. One of them are the complete bipartite graphs . For the other one, let denote the graph consisting of a path on vertices and additional vertices adjacent to all the vertices of the path. The following is straightforward, and also follows from a stronger Lemma 5 below.
Let be an integer. For every , there exists such that all -islands in and have more than vertices.
Thus a class with can contain only finitely many graphs of form or . We conjecture that for minor-closed classes, this condition is also sufficient.
A minor-closed class of graphs satisfies if and only if there exists such that and .
Conjecture 4 implies that (and hence ), since is a minor of and for all . In the next section we show that if Conjecture 4 holds, then for every minor-closed graph class . On the other hand, the parameters and are not tied, and thus, unfortunately, one can not hope to extend the methods of this paper to characterize minor-closed graph classes with given clustered chromatic number. We refer the reader to [21, Conjectures 30 and 32] for a conjectured characterization.
2 Our results
We start this section by presenting a strengthening of Observation 3 to clustered list chromatic number.
For all positive integers there exists a positive integer such that and .
We present the proof for , the proof for is similar. Since for every connected graph with more than vertices, we can assume that . Let be a set of size . Let be a -list assignment for such that for all and the vertices of are assigned disjoint subsets of . Suppose further that for every subset with there exists a set of consecutive vertices of such that for every . Clearly such a list assignment exists if is sufficiently large.
Consider an -coloring of . It remains to show that there exists a monochromatic connected subgraph of size more than . Let be the set of colors assigned to vertices of . Then by the choice of . Let for some color , and let be as defined in the previous paragraph. If some color in is used on at least vertices of then contains a monochromatic connected subgraph induced by these vertices and a vertex of . Otherwise, contains a monochromatic subpath of of length at least induced by the vertices in which are colored using color . ∎
Lemma 5 immediately implies the following.
Let be a class of graphs such that . Then there exists such that and .
Let denote the treewidth of the graph .
Let be a minor-closed class of graphs of bounded treewidth, and let be an integer. Then if and only if there exists such that and .
If is a minor-closed class of graphs of bounded treewidth, then .
Theorem 7 can be applied to bound the clustered chromatic number of minor-closed classes of unbounded treewidth. The key tool which allows such an application is the following theorem of DeVos et al.
Theorem 9 (DeVos et al. ).
For every minor-closed class there exists an integer , such that for every graph there exists a partition of satisfying and .
Let be a minor-closed class of graphs, and let be integers such that and . Then .
Our second main result bounds the clustered coloring number of minor-closed classes of graphs in terms of the maximum density of the class.
For every graph , integer and real there exists satisfying the following. Let be an -minor-free graph such that . Then contains a -island of size at most .
Theorem 11 immediately implies the following.
Let be a class of graphs closed under taking subgraphs, such that for some graph . Let real be such that for all graphs . Then .
For a surface let denote the (minor-closed) class of graphs embeddable on . Kleinberg, Motwani, Raghavan, and Venkatasubramanian  and, independently, Alon, Ding, Oporowski and Vertigan  proved that for every surface . Thus . By Euler’s formula, for every surface there exists a constant such that for all graphs . Thus Corollary 12 implies that the above lower bounds on and are tight, as conjectured by Esperet and Ochem [11, Conjecture 3].
For every surface
Let be the class of graphs with girth at least five which are embeddable on . The question of determining was considered, in particular, in , and was until now open. Euler’s formula again gives us a density bound for some and all , and thus Corollary 12 implies the following bound on conjectured in [11, Conjecture 7].
For every surface
Let be an integer. Then
Finally, let us discuss a relationship to another concept, bootstrap percolation. Consider the following process on a graph for some integer . Let vertices of some set be marked active. If there exists an inactive vertex in with at least active neighbors, becomes active. We repeat this procedure until there are no more inactive vertices with at least active neighbors. If at the end, all vertices of are active, we say that the set -percolates. Bootstrap percolation was introduced by Chalupa, Leath and Reich  as a simplification of existing models of ferromagnetism. Extremal problems for bootstrap percolation, similar to the ones we consider in this paper, were studied for very structured graph families e.g. in [4, 20].
For , we say that a graph is -resistant to -percolation if no set of at most vertices of -percolates. For a class of graphs , let us define the percolation threshold of to be the minimum integer such that for some , all non-null graphs in are -resistant to -percolation ( if no such exists). Clearly, all graphs in are also -resistant to -percolation for every .
How to show that a class is percolation resistant? Observe that a set -percolates if and only if no -island is contained in its complement. If a graph contains more than pairwise disjoint -islands, then each set of size at most is disjoint from at least one of them, and thus it does not -percolate. Hence, the following notion gives an upper bound to the percolation threshold. Let the pervasive clustered coloring number denote the minimum integer such that for some , each graph contains at least pairwise disjoint -islands ( if no such exists).
If contains linearly many pairwise disjoint -islands, some of them must have constant size. Hence, for a subgraph-closed class we have inequalities and . In general, percolation threshold may be smaller than the clustered coloring number; e.g., for the class of graphs with maximum degree at most and girth has clustered coloring number (since in -regular graphs in the class, all -islands must contain cycles) but all the graphs in this class are -resistant to -percolation (since the complements of sets of size at most contain a vertex of degree at most , or two vertices of degree joined by a path, or a cycle, forming a -island).
However, in Section 3 we prove the following.
Let be a class of graphs closed under taking subgraphs, such that for some graph . Then .
Hence for the minor closed classes percolation threshold bounds the clustered coloring number. Rather than proving Theorems 7 and 11 directly, we bound the percolation threshold and the pervasive clustered coloring number of the corresponding graph classes, proving the following strengthening of Theorems 7 and 11, respectively.
Let be a minor-closed class. If has bounded treewidth, then is equal to the smallest integer such that for some neither nor belongs to .
For every graph , integer and there exists satisfying the following. Let be an -minor-free graph satisfying . Then contains at least disjoint -islands.
Let be a class of graphs closed under taking subgraphs, such that for some graph . Let real be such that for all graphs . Then .
3 Percolation and clustered coloring in classes with sublinear separators
In this section we prove Theorems 16 and 18. In fact we show that these results hold not just for minor-closed classes, but for a wider family of graph classes which admit “good” separators. We start by defining this family.
A separation of a graph is a pair of its subgraphs such that ; note that is a vertex-cut in separating from , if these two sets are non-empty. Let be a non-decreasing function. We say that a class of graphs has -separators, if for every graph there exists a separation of of order at most such that . We say that a function is significantly sublinear if the sum
Let be a non-decreasing significantly sublinear function. Let be a class of graphs closed under taking subgraphs that has -separators. For every there exists as follows. For every -vertex graph there exists such that and every component of has at most vertices.
Since is significantly sublinear, there exists such that
Without loss of generality, we can assume that is connected, as otherwise we can find the set separately in each component. Let us define a rooted tree and a mapping of its vertices to connected induced subgraphs of as follows. For the root of , we set . For any , if , then is a leaf of . Otherwise, let be a separation of order at most and let . If , …, are the components of , then has children , …, in with for .
We let be the union of the sets over all non-leaf vertices of . By the construction of , every component of has at most vertices, and thus it suffices to bound the size of . For , define . Observe that the rank is decreasing on each path in starting in the root, and in particular, if for distinct , then and are vertex-disjoint. For any , let be the set of non-leaf vertices of of rank and let . Since for all , we have and , and thus . Consequently,
as required. ∎
In order to prove a strengthening of Theorem 18, we need one more definition. For a subset , let (or for ease of notation when the graph is clear from the context) denote the number of edges of with at least one end in . We say that is a -enclave if . The following observation is easy, but useful.
Let be a graph, and let be a -enclave in . Then there exists a -island in .
Choose a minimal -enclave . Note that , since . We claim that is an island, as desired. Suppose for a contradiction that there exists with at least neighbors in . Then
Thus is a -enclave, contradicting the choice of . ∎
Let be a non-decreasing significantly sublinear function. Let be a class of graphs closed under taking subgraphs that has -separators. For every integer and there exists so that every satisfying contains at least disjoint -islands.
Let . Let be chosen to satisfy the conclusion of Lemma 20 for this and , and let . Consider an -vertex graph satisfying . By the choice of there exists a set of size at most such that every component of has at most vertices. Let be the collection of vertex sets of the components of , let be the collection of the sets in which are -enclaves, and let . By Lemma 21 it suffices to show that . If not, then
Moreover, we have for every , and thus
which contradicts the assumption of the theorem. ∎
Let us remark that the argument of Theorem 22 also directly implies Theorem 11, since all the -enclaves we obtain have bounded size (it is possible to simplify the proof a bit in this weakened case, since we only need to prove the existence of one such -enclave).
The following result with an analogous proof implies Theorem 16.
Let be a non-decreasing significantly sublinear function. Let be a class of graphs closed under taking subgraphs that has -separators. Then .
Since in general, it suffices to show that . Letting , there exists such that is -resistant to -percolation. Let be chosen to satisfy the conclusion of Lemma 20 for and . We claim that every graph contains at least disjoint -islands. The claim implies that , and hence the theorem.
It remains to establish the claim. Let be an -vertex graph. By the choice of , there exists such that and every component of has at most vertices. Let be the set of all components of such that there exists a -island in . It suffices to show that . If not, then the set has size at most . By the choice of there exists a -island . Choose such to be minimal, then is connected, and so for some component of . However, by the choice of . This contradiction finishes the proof. ∎
4 Clustered chromatic number of classes of bounded treewidth
We start by introducing the necessary concepts. For sets of vertices of a graph of the same size , an linkage is a set of pairwise vertex-disjoint paths with one end in and the other end in .
Let and be graphs. An -decomposition of is a function that to each vertex of assigns a subset of vertices of (called the bag of ), such that for every , there exists with , and for every , the set induces a non-empty connected subgraph of . When is a path or a tree, we say that is a path or tree decomposition of , respectively. The width of an -decomposition is defined as , and the adhesion of an -decomposition is . The treewidth of a graph is the minimum width of a tree decomposition of .
A path decomposition is linked if for every with two neighbors and , there exists a linkage in (and in particular, all the intersections of adjacent bags have the same size). Conversely, if and there exists a separation of of order less than with and , we say that has broken bag; by Menger’s theorem, a path decomposition such that all the intersections of adjacent bags have the same size is linked if and only if it does not contain a broken bag. An -decomposition is proper if for every . The order of an -decomposition is . For a path decomposition , a path decomposition is a coarsening of if there exists a model of in such that and for all . A path decomposition of is appearance-universal if every vertex either appears in all bags of the decomposition, or in at most two (consecutive) bags. A vertex is internal if it appears in only one bag. A path decomposition has large interiors if for every with two neighbors and , there exists an internal vertex contained in and no internal vertex has a neighbor both in and in .
We now show how to transform a tree decomposition of large order and bounded width into a path decomposition of large order and bounded adhesion, and then to clean it up, making it linked, appearance-universal, and with large interiors. The techniques to do so are standard and appear e.g. in ; we give brief arguments here to adjust for minor technical details and notational differences.
A coarsening of a proper decomposition is proper. A coarsening of a linked decomposition is linked. A coarsening of a decomposition of adhesion has adhesion at most . A coarsening of an appearance-universal decomposition is appearance-universal; furthermore, if an appearance-universal decomposition has large interiors, then its coarsening has large interiors.
Let be integers. If a graph has a proper tree decomposition of order at least with bags of size at most , then has a proper path decomposition of adhesion at most and order at least .
If has a vertex of degree , then let , …, be the components of , let be a path and let . Otherwise, contains a subpath with at least vertices; for , let be the component of containing , and let . In both cases, is a proper path decomposition of of adhesion at most . ∎
There exists a function as follows. Let and be integers. If has a proper path decomposition with adhesion at most of order at least , then has a proper linked path decomposition of adhesion at most and order .
Choose so that and for every . We prove the claim by the induction on .
If , then is linked and the claim holds trivially. If there exist at least edges such that , then has a coarsening with adhesion at most and order at least , and the claim follows by the induction hypothesis. Hence, assume that there are at most such edges, and thus there exists a coarsening of of order such that any two adjacent bags intersect in exactly vertices. If there exist at least vertices of with broken bags, then has a proper path decomposition with adhesion at most of order at least , and the claim follows by induction. Otherwise, since has order greater than , it contains consecutive vertices with unbroken bags, and thus it has a coarsening of order in that no bag is broken. This coarsening is a proper linked path decomposition of . ∎
Let and be integers. Every path decomposition of a graph with adhesion at most of order at least has an appearance-universal coarsening of order .
We prove the claim by the induction on . If , then every vertex appears in exactly one bag and is appearance-universal. If a vertex appears in at least bags of the decomposition, then there exists a coarsening of order such that appears in all the bags. Let for all . Then is a path decomposition of of adhesion at most , and by the induction hypothesis, it has a coarsening of order that is appearance-universal. Let for all . Then is an appearance-universal coarsening of of order .
Hence, we can assume that every vertex appears in at most bags of the decomposition. Let be the coarsening obtained by dividing into subpaths with vertices (plus possibly one shorter path at the end) and merging the bags in the subpaths. Then every vertex appears in at most two consecutive bags, and thus is appearance-universal. ∎
Let be an integer. Let . Every proper appearance-universal path decomposition of order at least has a coarsening of order with large interiors.
Divide into subpaths with three vertices and merge the bags in each subpath, obtaining a coarsening of order . We claim that this coarsening has large interiors.
Indeed, suppose that is a subpath of and consider a vertex such that . If a vertex belongs both to and , then also , and since is appearance-universal, it follows that belongs to all its bags. Similarly, any vertex in belongs to all bags. Since is proper, some vertex of does not belong to all bags, and thus it is internal in .
Let and be the neighbors of in . Consider now an internal vertex . Since is internal in , it does not appear in all bags of , and by appearance-universality, appears in at most two consecutive bags of . Consequently, does not appear in the bags of both and , and by symmetry, we can assume that . Hence, any neighbor of in must belong to , and by appearance-universality it must belong to all bags. It follows that does not have a neighbor both in and in . ∎
A -tuple , where is a graph, is a union of pairwise vertex-disjoint paths in , and are injective functions such that for , and are the two endpoints of one of the paths of , is called an extended bag of adhesion . Let be a linked path decomposition of a graph . For each with two neighbors and , fix a linkage in . Let be the path obtained from by removing its endpoints. Let be the union of the linkages over all . Note that is the disjoint union of paths , …, . Order the path arbitrarily, and let be a vertex of whose predecessor in is and whose successor in is . Let and be functions defined so that is the vertex of belonging to and is the vertex of belonging to . The 4-tuple is the extended bag of . A finite extended bag property is a function that to each extended bag of adhesion assigns an element of a finite set . By the pigeonhole principle, we have the following.
For all integers and and for any finite extended bag property , there exists an integer as follows. For any linked decomposition of adhesion at most and any such that , there exists a set of size such that for all .
We are now ready to start working towards the proof of Theorem 17. Let us start with a lemma that enables us to obtain many -islands separated from the rest of the graph by small cuts.
For all integers and , there exists an integer as follows. Let be a linked path decomposition of a graph , with large interiors, adhesion at most and order at least . Then either contains or as a minor, or there exists a subpath of of length such that the internal vertices of the bag of form a -island in for every .
Suppose that there does not exist a subpath of as above. Let be the set of vertices of of degree two such that the internal vertices of the bag of do not form a -island in . Then . Consider any vertex of degree two, and let be its extended bag. Since the internal vertices of the bag of do not form a -island, there exists an internal vertex with at least non-internal neighbors. Choose of the non-internal neighbors , …, , and let , where for , we have if or , and if lies on the path in with ends and , and otherwise. Observe that , …, are distinct, since has large interiors. Note that is a finite extended bag property, and by Observation 29, we can assume that there exist distinct vertices , …, in