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Clique-width and Well-Quasi-Ordering of Triangle-Free Graph Classes

11/23/2017
by   Konrad K. Dabrowski, et al.
Durham University
University of Warwick
0

Daligault, Rao and Thomassé asked whether every hereditary graph class that is well-quasi-ordered by the induced subgraph relation has bounded clique-width. Lozin, Razgon and Zamaraev (JCTB 2017+) gave a negative answer to this question, but their counterexample is a class that can only be characterised by infinitely many forbidden induced subgraphs. This raises the issue of whether the question has a positive answer for finitely defined hereditary graph classes. Apart from two stubborn cases, this has been confirmed when at most two induced subgraphs H_1,H_2 are forbidden. We confirm it for one of the two stubborn cases, namely for the (H_1,H_2)=(triangle,P_2+P_4) case, by proving that the class of (triangle,P_2+P_4)-free graphs has bounded clique-width and is well-quasi-ordered. Our technique is based on a special decomposition of 3-partite graphs. We also use this technique to prove that the class of (triangle,P_1+P_5)-free graphs, which is known to have bounded clique-width, is well-quasi-ordered. Our results enable us to complete the classification of graphs H for which the class of (triangle,H)-free graphs is well-quasi-ordered.

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1 Introduction

A graph class  is well-quasi-ordered by some containment relation if for any infinite sequence of graphs in , there is a pair with such that  is contained in . A graph class  has bounded clique-width if there exists a constant  such that every graph in  has clique-width at most . Both being well-quasi-ordered and having bounded clique-width are highly desirable properties of graph classes in the areas of discrete mathematics and theoretical computer science. To illustrate this, let us mention the seminal project of Robertson and Seymour on graph minors that culminated in 2004 in the proof of Wagner’s conjecture, which states that the set of all finite graphs is well-quasi-ordered by the minor relation. As an algorithmic consequence, given a minor-closed graph class, it is possible to test in cubic time whether a given graph belongs to this class. The algorithmic importance of having bounded clique-width follows from the fact that many well-known NP-hard problems, such as Graph Colouring and Hamilton Cycle, become polynomial-time solvable for graph classes of bounded clique-width (this follows from combining results from several papers [6, 16, 19, 27] with a result of Oum and Seymour [25]).

Courcelle [5] proved that the class of graphs obtained from graphs of clique-width  via one or more edge contractions has unbounded clique-width. Hence the clique-width of a graph can be much smaller than the clique-width of its minors. On the other hand, the clique-width of a graph is at least the clique-width of any of its induced subgraphs (see, for example, [7]). We therefore focus on hereditary classes, that is, on graph classes that are closed under taking induced subgraphs. It is readily seen that a class of graphs is hereditary if and only if it can be characterised by a unique set  of minimal forbidden induced subgraphs. Our underlying research goal is to increase our understanding of the relation between well-quasi-orders and clique-width of hereditary graph classes.

As a start, we note that the hereditary class of graphs of degree at most  is not well-quasi-ordered by the induced subgraph relation, as it contains the class of cycles, which form an infinite antichain. As every graph of degree at most  has clique-width at most , having bounded clique-width does not imply being well-quasi-ordered by the induced subgraph relation. In 2010, Daligault, Rao and Thomassé [13] asked about the reverse implication: does every hereditary graph class that is well-quasi-ordered by the induced subgraph relation have bounded clique-width? In 2015, Lozin, Razgon and Zamaraev [24] gave a negative answer. As the set  of minimal forbidden induced subgraphs in their counter-example is infinite, the question of Daligault, Rao and Thomassé [13] remains open for finitely defined hereditary graph classes, that is, hereditary graph classes for which  is finite.

Conjecture 1 ([24])

If a finitely defined hereditary class of graphs  is well-quasi-ordered by the induced subgraph relation, then  has bounded clique-width.

If Conjecture 1 is true, then for finitely defined hereditary graph classes the aforementioned algorithmic consequences of having bounded clique-width also hold for the property of being well-quasi-ordered by the induced subgraph relation. A hereditary graph class defined by a single forbidden induced subgraph  is well-quasi-ordered by the induced subgraph relation if and only if it has bounded clique-width if and only if  is an induced subgraph of  (see, for instance, [12, 14, 21]). Hence Conjecture 1 holds when  has size . We consider the case when  has size , say . Such graph classes are said to be bigenic or -free graph classes. In this case Conjecture 1 is also known to be true except for two stubborn open cases, namely and ; see [10].

Our Results. In Section 4, we prove that Conjecture 1 holds for the class of -free graphs by showing that the class of -free graphs has bounded clique-width and is well-quasi-ordered by the induced subgraph relation. We do this by using a general technique explained in Section 3. This technique is based on extending (a labelled version of) well-quasi-orderability or boundedness of clique-width of the bipartite graphs in a hereditary graph class  to a special subclass of -partite graphs in . The crucial property of these -partite graphs is that no three vertices from the three different partition classes form a clique or independent set. We say that such -partite graphs curious. A more restricted version of this concept was used to prove that -free graphs have bounded clique-width [8]. In Section 4 we show how to generalise results for curious -free graphs to the whole class of -free graphs. In the same section we also show how to apply our technique to prove that the class of -free graphs is well-quasi-ordered (it was already known [8] that the class of -free graphs has bounded clique-width). We note that our results also imply that the class of -free graphs is well-quasi-ordered, which was not previously known (see [2, 21]). See also Figure 1 for pictures of the forbidden induced subgraphs mentioned in this paragraph.

Figure 1: The forbidden induced subgraphs considered in our results.

Dichotomies. Previously, well-quasi-orderability was known for -free graphs [2], -free bipartite graphs [20] and -free bipartite graphs [20]. It has also been shown that -free bipartite graphs are not well-quasi-ordered if  contains an induced  [21],  [15] or  [20]. Moreover, for every , the class of -free graphs is finite due to Ramsey’s Theorem [26]. The above results lead to the following known dichotomy for -free bipartite graphs.

Theorem 1.1

Let  be a graph. The class of -free bipartite graphs is well-quasi-ordered by the induced subgraph relation if and only if for some or  is an induced subgraph of , or .

Now combining the aforementioned known results for -free graphs and -free bipartite graphs with our new results yields the following new dichotomy for -free triangle-free graphs, which is exactly the same as the one in Theorem 1.1.

Theorem 1.2

Let  be a graph. The class of -free graphs is well-quasi-ordered by the induced subgraph relation if and only if for some or is an induced subgraph of , , or .

Besides our technique based on curious graphs, we also expect that Theorem 1.2 will itself be a useful ingredient for showing results for other graph classes, just as Theorem 1.1 has already proven to be useful (see e.g. [20]).

For clique-width the following dichotomy is known for -free bipartite graphs.

Theorem 1.3 ([11])

Let  be a graph. The class of -free bipartite graphs has bounded clique-width if and only if for some or  is an induced subgraph of , , or .

It would be interesting to determine whether -free graphs allow the same dichotomy with respect to the boundedness of their clique-width. The evidence so far is affirmative, but in order to answer this question two remaining cases need to be solved, namely and ; see Section 2 for the definition of the graph . Both cases turn out to be highly non-trivial; in particular, the class of -free graphs contains the class of -free graphs, and the class of -free graphs contains both the classes of -free and -free graphs.

In Section 5 we give state-of-the-art summaries for well-quasi-orderability and boundedness of clique-width of bigenic graph classes (which include our new results), together with an overview of the missing cases for both problems (which include the missing cases mentioned in this section).

2 Preliminaries

Throughout the paper, we consider only finite, undirected graphs without multiple edges or self-loops. Below, we define further graph terminology.

The disjoint union of two vertex-disjoint graphs  and  is denoted by  and the disjoint union of  copies of a graph  is denoted by . The complement  of a graph  has vertex set and an edge between two distinct vertices if and only if . For a subset , we let  denote the subgraph of  induced by , which has vertex set  and edge set . If then, to simplify notation, we may also write instead of . We use to denote the graph obtained from  by deleting every vertex in , that is, . We write to indicate that  is an induced subgraph of .

The graphs , , and  denote the cycle, complete graph, star and path on  vertices, respectively. The graphs  and  are also called the triangle and claw, respectively. A graph  is a linear forest if every component of  is a path (on at least one vertex). The graph , for , denotes the subdivided claw, that is, the tree that has only one vertex  of degree  and exactly three leaves, which are of distance  and  from , respectively. Observe that . We let  denote the class of graphs, each connected component of which is either a subdivided claw or a path.

For a set of graphs , a graph  is -free if it has no induced subgraph isomorphic to a graph in ; if , we may write -free instead of -free. A graph is -partite if its vertex can be partitioned into  (possibly empty) independent sets; -partite graphs are also known as bipartite graphs. The biclique or complete bipartite graph  is the bipartite graph with sets in the partition of size  and  respectively, such that every vertex in one set is adjacent to every vertex in the other set. For a graph , the set denotes the neighbourhood of . Let  be a set of vertices in . A vertex is complete to  if it is adjacent to every vertex of  and anti-complete to  if it is non-adjacent to every vertex of . A set of vertices is complete (resp. anti-complete) to  if every vertex in  is complete (resp. anti-complete) to . If  and  are disjoint sets of vertices in a graph, we say that the edges between these two sets form a matching if each vertex in  has at most one neighbour in  and vice versa (if each vertex has exactly one such neighbour, we say that the matching is perfect). Similarly, the edges between these sets form a co-matching if each vertex in  has at most one non-neighbour in  and vice versa. We say that the set  dominates  if every vertex of  has at least one neighbour in . Similarly, a vertex  dominates  if every vertex of  is adjacent to . A vertex distinguishes  if  has both a neighbour and a non-neighbour in . The set  is a module of  if no vertex in distinguishes . A module  is non-trivial if , otherwise it is trivial. A graph is prime if it has only trivial modules. Two vertices are false twins if they have the same neighbourhood (note that such vertices must be non-adjacent). Clearly any prime graph on at least three vertices cannot contain a pair of false twins, as any such pair of vertices would form a non-trivial module.

We will use the following structural result.

Lemma 1 ([8])

Let  be a connected -free graph that does not contain a pair of false twins. Then  is either bipartite or a cycle.

2.1 Clique-width

The clique-width  of a graph  is the minimum number of labels needed to construct  by using the following four operations:

  1. : creating a new graph consisting of a single vertex  with label ;

  2. : taking the disjoint union of two labelled graphs  and ;

  3. : joining each vertex with label  to each vertex with label  ();

  4. : renaming label  to .

A class of graphs  has bounded clique-width if there is a constant  such that the clique-width of every graph in  is at most ; otherwise the clique-width is unbounded.

Let  be a graph. We define the following operations. For an induced subgraph , the subgraph complementation operation (acting on  with respect to ) replaces every edge present in  by a non-edge, and vice versa. Similarly, for two disjoint vertex subsets  and  in , the bipartite complementation operation with respect to  and  acts on  by replacing every edge with one end-vertex in  and the other one in  by a non-edge and vice versa.

We now state some useful facts about how the above operations (and some other ones) influence the clique-width of a graph. We will use these facts throughout the paper. Let be a constant and let  be some graph operation. We say that a graph class  is -obtained from a graph class  if the following two conditions hold:

  1. every graph in  is obtained from a graph in  by performing  at most  times, and

  2. for every there exists at least one graph in  obtained from  by performing  at most  times.

We say that  preserves boundedness of clique-width if for any finite constant  and any graph class , any graph class  that is -obtained from  has bounded clique-width if and only if  has bounded clique-width.

  1. Vertex deletion preserves boundedness of clique-width [23].

  2. Subgraph complementation preserves boundedness of clique-width [18].

  3. Bipartite complementation preserves boundedness of clique-width [18].

Lemma 2 ([7])

Let  be a graph and let  be the set of all induced subgraphs of  that are prime. Then .

2.2 Well-quasi-orderability

A quasi order  on a set  is a reflexive, transitive binary relation. Two elements in this quasi-order are comparable if or , otherwise they are incomparable. A set of elements in a quasi-order is a chain if every pair of elements is comparable and it is an antichain if every pair of elements is incomparable. The quasi-order  is a well-quasi-order if any infinite sequence of elements in  contains a pair with and . Equivalently, a quasi-order is a well-quasi-order if and only if it has no infinite strictly decreasing sequence and no infinite antichain.

For an arbitrary set , let  denote the set of finite sequences of elements of . A quasi-order  on  defines a quasi-order  on  as follows: if and only if there is a sequence of integers with such that for . We call  the subsequence relation.

Lemma 3 (Higman’s Lemma [17])

If is a well-quasi-order then is a well-quasi-order.

To define the notion of labelled induced subgraphs, let us consider an arbitrary quasi-order . We say that  is a labelled graph if each vertex  of  is equipped with an element (the label of ). Given two labelled graphs  and , we say that  is a labelled induced subgraph of  if  is isomorphic to an induced subgraph of  and there is an isomorphism that maps each vertex  of  to a vertex  of  with . Clearly, if is a well-quasi-order, then a graph class  cannot contain an infinite sequence of labelled graphs that is strictly-decreasing with respect to the labelled induced subgraph relation. We therefore say that a graph class  is well-quasi-ordered by the labelled induced subgraph relation if it contains no infinite antichains of labelled graphs whenever is a well-quasi-order. Such a class is readily seen to also be well-quasi-ordered by the induced subgraph relation.

Daligault, Rao and Thomassé [13] showed that every hereditary class of graphs that is well-quasi-ordered by the labelled induced subgraph relation is defined by a finite set of forbidden induced subgraphs. Korpelainen, Lozin and Razgon [22] conjectured that if a hereditary class of graphs  is defined by a finite set of forbidden induced subgraphs, then  is well-quasi-ordered by the induced subgraph relation if and only if it is well-quasi-ordered by the labelled induced subgraph relation. Brignall, Engen and Vatter [4] recently found a class  with 14 forbidden induced subgraphs that is a counterexample for this conjecture, that is is well-quasi-ordered by the induced subgraph relation but not by the labelled induced subgraph relation. However, so far, all known results for bigenic graph classes, including those in this paper, verify the conjecture for bigenic graph classes.

Similarly to the notion of preserving boundedness of clique-width, we say that a graph operation  preserves well-quasi-orderability by the labelled induced subgraph relation if for any finite constant  and any graph class , any graph class  that is -obtained from  is well-quasi-ordered by this relation if and only if  is.

Lemma 4

[10] The following operations preserve well-quasi-orderability by the labelled induced subgraph relation:

  1. Subgraph complementation,

  2. Bipartite complementation and

  3. Vertex deletion.

Lemma 5 ([2])

A hereditary class  of graphs is well-quasi-ordered by the labelled induced subgraph relation if and only if the set of prime graphs in  is. In particular, is well-quasi-ordered by the labelled induced subgraph relation if and only if the set of connected graphs in  is.

Lemma 6 ([2, 20])

-free bipartite graphs are well-quasi-ordered by the labelled induced subgraph relation.

Let  and  be well-quasi-orders. We define the Cartesian Product of these well-quasi-orders as the order on the set where if and only if and . Lemma 3 implies that is also a well-quasi-order. If  has a labelling with elements of  and of , say and , we can construct the combined labelling in that labels each vertex  of  with the label .

Lemma 7

Fix a well-quasi-order  that has at least one element. Let  be a class of graphs. For each fix a labelling . Then  is well-quasi-ordered by the labelled induced subgraph relation if and only if for every well-quasi-order  and every labelling of the graphs in  by this order, the combined labelling in obtained from these labellings also results in a well-quasi-ordered set of labelled graphs.

Proof

If  is well-quasi-ordered by the labelled induced subgraph relation then by definition it is well-quasi-ordered when labelled with labels from these combined labellings obtained from these well-quasi-orders. If  is not well-quasi-ordered by the labelled induced subgraph relation then there must be a well-quasi-order and an infinite set of graphs whose vertices are labelled with elements of  such that these graphs form an infinite labelled antichain. For each graph , replace the label  on vertex  by . The graphs are now labelled with elements of the well-quasi-order and result in an infinite labelled antichain of graphs labelled with such combined labellings. This completes the proof.∎

2.3 -uniform Graphs

For an integer , a graph  is -uniform if there is a symmetric square matrix  of order  and a graph  on vertices such that , where is the graph class defined as follows. Let  be the disjoint union of infinitely many copies of . For , let  be the subset of  containing vertex  from each copy of . Construct from  an infinite graph  on the same vertex set by applying a subgraph complementation to  if and only if and by applying a bipartite complementation to a pair if and only if . Thus, two vertices and are adjacent in  if and only if and or and . Then, is the hereditary class consisting of all the finite induced subgraphs of . The minimum  such that  is -uniform is the uniformicity of . The second of the next two lemmas follows directly from the above definitions.

The following result was proved by Korpelainen and Lozin. The class of disjoint unions of cliques is a counterexample for the reverse implication.

Lemma 8 ([21])

Any class of graphs of bounded uniformicity is well-quasi-ordered by the labelled induced subgraph relation.

The following lemma follows directly from the definition of clique-width and the definition of -uniform graphs (see also [1] for a more general result).

Lemma 9

Every -uniform graph has clique-width at most .

3 Partitioning -Partite Graphs

In Section 3.1, we first introduce a graph decomposition on -partite graphs. We then show how to extend results on bounded clique-width or well-quasi-orderability by the labelled induced subgraph relation from bipartite graphs in an arbitrary hereditary class of graphs to the -partite graphs in this class that are decomposable in this way. In Section 3.2, we then give sufficient conditions for a -partite graph to have such a decomposition.

3.1 The Decomposition

Let  be a -partite graph given with a partition of its vertex set into three independent sets , and . Suppose that each set  can be partitioned into sets  such that, taking subscripts modulo :

  • for if then  is complete to  and anti-complete to .

For let . We say that the graphs  are the slices of . If the slices belong to some graph class , then we say that  can be partitioned into slices from ; see Figure 2 for an example.

Figure 2: A -partite graph partitioned into slices isomorphic to .
Lemma 10

If  is a -partite graph that can be partitioned into slices of clique-width at most  then  has clique-width at most .

Proof

Since every slice  of  has clique-width at most , it can be constructed using the labels . Applying relabelling operations if necessary, we may assume that at the end of this construction, every vertex receives the label . We can modify this construction so that we use the labels instead, in such a way that at all points in the construction, for each every constructed vertex in  has a label in . To do this we replace:

  • creation operations  by  if ,

  • relabel operations  by and

  • join operations  by
           .

This modified construction uses  labels and at the end of it, every vertex in  is labelled with label . We may do this for every slice  of  independently. We now show how to use these constructed slices to construct using six labels in such a way that every vertex in  is labelled with label . We do this by induction. If then , so we are done. If then by the induction hypothesis, we can construct in this way. Consider the copy of  constructed earlier and relabel its vertices using the operations , and so that in this copy of , every vertex in  is labelled . Next take the disjoint union of the obtained graph with the constructed . Then, apply join operations , and . Finally, apply the relabelling operations , and . This constructs in such a way that every vertex in  is labelled with . By induction, it follows that  has clique-width at most .∎

Lemma 11

Let  be a hereditary graph class containing a class . Let  be the set of -partite graphs in  that can be partitioned into slices from . If  is well-quasi-ordered by the labelled induced subgraph relation then so is .

Proof

For each graph  in , we may fix a partition into independent sets with respect to which the graph can be partitioned into slices from . Let be the well-quasi-order with in which every pair of distinct elements is incomparable. By Lemma 7, we need only consider labellings of graphs in  of the form where and  belongs to an arbitrary well-quasi-order . Suppose  can be partitioned into slices , with vertices labelled as in . The slices along with the labellings completely describe the edges in . Suppose  is another such graph, partitioned into slices . If is smaller than under the subsequence relation, then  is an induced subgraph of . The result follows by Lemma 3.∎

3.2 Curious Graphs

Let  be a -partite graph given together with a partition of its vertex set into three independent sets , and . An induced  or  in  is rainbow if it has exactly one vertex in each set . We say that  is curious with respect to the partition if it contains no rainbow  or  when its vertex set is partitioned in this way. We say that  is curious if there is a partition with respect to which  is curious. In this section we will prove that given a hereditary class , if the bipartite graphs in  are well-quasi-ordered by the labelled induced subgraph relation or have bounded clique-width, then the same is true for the curious graphs in .

A linear order of the vertices of an independent set  is

  • increasing if implies ,

  • decreasing if implies ,

  • monotone if it is either increasing or decreasing.

Bipartite graphs that are -free are also known as bipartite chain graphs. It is and it is well known (and easy to verify) that a bipartite graph  is -free if and only if the vertices in each independent set of the bipartition admit a monotone ordering. Suppose  is a curious graph with respect to some partition . We say that (with respect to this partition) the graph  is a curious graph of type  if exactly  of the graphs , and  contain an induced .

3.2.1 Curious Graphs of Type  and .

Note that if  is a curious graph of type  or  with respect to the partition then without loss of generality, we may assume that and are both -free.

Lemma 12

Let  be a curious graph with respect to , such that and are both -free. Then the vertices of  admit a linear ordering which is decreasing in and increasing in .

Proof

For a set , we use to denote the set of vertices in  that are adjacent to . We may choose a linear order of the vertices of  according to their neighbourhood in , breaking ties according to their neighbourhood in  i.e. an order such that:

  1. if then and

  2. if and then .

Clearly such an ordering is decreasing in .

Suppose, for contradiction, that this order is not increasing in . Then there must be indices such that . Then by Property (ii). By Property (i) it follows that . This means that there are vertices and . Now if  is adjacent to  then is a rainbow  and if  is non-adjacent to  then is a rainbow . This contradiction implies that the order is indeed decreasing in , which completes the proof.∎

Lemma 13

If  is a curious graph of type  or  with respect to a partition then  can be partitioned into slices that are bipartite.

Proof

Let be a linear order on  satisfying Lemma 12. Let and for , let . We partition  and  as follows. For , let . For , let . In particular, note that the vertices of and  are complete and anti-complete to , respectively. The following properties hold:

  • If then  is complete to  and anti-complete to .

  • If then  is anti-complete to  and complete to .

If and is non-adjacent to then is a rainbow , a contradiction. If and is adjacent to then is a rainbow , a contradiction. It follows that:

  • If then  is complete to .

  • If then  is anti-complete to .

For , let . The above properties about the edges between the sets  show that  can be partitioned into the slices . Now, for each , is anti-complete to , so every slice  is bipartite. This completes the proof.∎

3.2.2 Curious Graphs of Type  and .

Lemma 14

Fix . If  is a curious graph of type  with respect to a partition then  can be partitioned into slices of type at most .

Proof

Fix and let  be a curious graph of type  with respect to a partition . We may assume that contains an induced .

Claim 1. Given a  in , every vertex of  has exactly two neighbours in the  and these neighbours either both lie in  or both lie in .
Let and induce a  in  such that  is adjacent to  but not to  and  is adjacent to  but not to . Consider . For , if  is adjacent to both  and , then is a rainbow , a contradiction. Therefore  can have at most one neighbour in and at most one neighbour in . If  is non-adjacent to  and  then is a rainbow , a contradiction. Therefore, can have at most one non-neighbour in , and similarly  have have at most one non-neighbour in . Therefore, if  is adjacent to , then it must be non-adjacent to , so it must be adjacent to , so it must be non-adjacent to . Similarly, if  is non-adjacent to  then it must be adjacent to , so it must be non-adjacent to , so it must be adjacent to . Hence Claim 3.2.2 follows.

Consider a maximal set of vertex-disjoint sets that induce copies of  in . We say that a vertex of  distinguishes two graphs  and  if its neighbours in  and  do not belong to the same set . We group these sets  into blocks that are not distinguished by any vertex of . In other words, every  contains at least one  and every vertex of  is complete to one of the sets and and anti-complete to the other. For , let . We define a relation  on the blocks as follows:

  • holds if  is complete to , while  is anti-complete to .

For distinct blocks , at most one of and can hold.

Claim 2. Let  and  be distinct blocks. There is a vertex that differentiates  and . If  is complete to and anti-complete to then (see also Figure 3). If  is complete to and anti-complete to then .
By definition of the blocks  and  there must be a vertex that distinguishes them. Without loss of generality we may assume that  is complete to and anti-complete to . It remains to show that . If is adjacent to then is a rainbow , a contradiction. If is non-adjacent to then is a rainbow , a contradiction. Therefore  is anti-complete to  and  is complete to . It follows that . Claim 3.2.2 follows by symmetry.

Figure 3: Two blocks  and  with and a vertex differentiating them.

Claim 3. The relation  is transitive.
Suppose that and . Since  and  are distinct, there must be a vertex that distinguishes them and combining Claim 3.2.2 with the fact that it follows that  must be complete to . Suppose and . Then  is adjacent to , since and  is adjacent to  by choice of . Now  is must be non-adjacent to  otherwise would be a rainbow . Therefore  is anti-complete to , so  distinguishes  and . By Claim 3.2.2 it follows that . This completes the proof of Claim 3.2.2.

Combining Claims 3.2.23.2.2, we find that  is a linear order on the blocks. We obtain the following conclusion, which we call the chain property.

Claim 4. The set of blocks admits a linear order such that

  1. if  then  is complete to , while  is anti-complete to  and

  2. for each there exists an such that if then  is complete to  and anti-complete to  and if then  is anti-complete to  and complete to .

Next consider the set of vertices in that do not belong to any set . Let  denote this set and note that  is -free by maximality of the set . For let