On clique-colouring of graphs with few P₄’s

Abstract

Let G=(V,E) be a graph with n vertices. A clique-colouring of a graph is a colouring of its vertices such that no maximal clique of size at least two is monocoloured. A k-clique-colouring is a clique-colouring that uses k colours. The clique-chromatic number of a graph G is the minimum k such that G has a k-clique-colouring.

In this paper we will use the primeval decomposition technique to find the clique-chromatic number and the clique-colouring of well known classes of graphs that in some local sense contain few P₄’s. In particular we shall consider the classes of extended P₄-laden graphs, p-trees (graphs which contain exactly n−3 P₄’s) and (q,q−3)-graphs, q≥7, such that no set of at most q vertices induces more that q−3 distincts P₄’s. As corollary we shall derive the clique-chromatic number and the clique-colouring of the classes of cographs, P₄-reducible graphs, P₄-sparse graphs, extended P₄-reducible graphs, extended P₄-sparse graphs, P₄-extendible graphs, P₄-lite graphs, P₄-tidy graphs and P₄-laden graphs that are included in the class of extended P₄-laden graphs.

Introduction

In this paper we are concerned with the so called clique-colouring of a graph. To introduce this concept we need the following definitions. A subset K of G is a clique if every pair of distinct vertices of K are adjacent in G. A clique is maximal if it is not properly contained in any other clique of G. A hypergraph is a pair $\mathcal{H}$$=(V, \mathcal{E})$, where V is the set of vertices of $\mathcal{H}$ and $\mathcal{E}$ is a family of non-empty subsets of V called edges. A k-colouring of $\mathcal{H}$ is a mapping c:V→{1,2,…,k} such that for all $e \in\mathcal {E}$ with |e|≥2 there exist u,v∈e with c(u)≠c(v), that is, there is no monocoloured edge of size at least two. The chromatic number$\chi( \mathcal{H})$ of $\mathcal{H}$ is the smallest k such that $\mathcal{H}$ has a k-colouring.

We will consider special hypergraphs: hypergraphs arising from graphs. Given a graph G=(V,E), the clique-hypergraph of G is the hypergraph $\mathcal{H} (G)$$= (V, \mathcal{E})$, whose vertices are the vertices of G and whose edges are the maximal cliques of G. A k-colouring of $\mathcal{H} (G)$ will also be called a k-clique-colouring of G and the chromatic number$\chi( \mathcal{H})$ of $\mathcal{H}$ the clique-chromatic number of G. In other words: a clique-colouring of a graph is a colouring of its vertices such that no maximal clique of size at least two is monocoloured.

Clique-colouring is harder than ordinary vertex colouring: it is coNP-complete even to check whether a 2-clique-colouring is valid [7]. The complexity of 2-clique-colourability is investigated in [26] where they show it is NP-hard to decide whether a perfect graph is 2-clique-colourable even for those with clique-number 3. A valid 2-clique-colouring is not a good certificate, since we cannot verify it in polynomial time. In [27] it is proved that it is $\varSigma_{2}^{p}$-complete to check whether a graph is 2-clique-colourable, even for odd-hole-free graphs [15]. However quite general classes of graphs have been proved to be 2-clique colourable or 3-clique colourable. In [7] it was proved that K_1,3-free graphs are 2-clique colourable and that almost all perfect graphs are 3-clique colourable. In [29] it was shown that every planar graph is 3-clique colourable and in [26] it was proposed a polynomial algorithm to decide if a planar graph is 2-clique colourable. The clique-colourability of several other classes of graphs has been investigated in [1, 9, 10, 14, 19].

In this paper we consider some classes of graphs that have been characterized in terms of special properties of the unique primeval decomposition tree associated to each graph of the class. The primeval decomposition tree of any graph can be computed in linear time [8] and therefore it is the natural framework for finding polynomial time algorithms of many problems.

Using the primeval decomposition technique, we will determine the clique-chromatic number of graphs with few P₄’s. In particular we shall consider the classes of extended P₄-laden [16], p-trees [3] (graphs which contain exactly n−3 P₄’s) and (q,q−3)-graphs, q≥7, [2] such that no set of at most q vertices induces more that q−3 distincts P₄’s. As corollary we shall derive the clique-chromatic number and the clique-colouring of the classes of cograph [11, 12], P₄-reducible graphs [22], P₄-sparse graphs [20, 24], extended P₄-reducible graphs [17], extended P₄-sparse graphs [17], P₄-extendible graphs [23], P₄-lite graphs [21], P₄-tidy graphs [18] and P₄-laden graphs [16] that are included in the class of extended P₄-laden graphs. Furthermore we will extend these results to more general classes obtained by substituting vertices of the above classes by homogeneous sets.

In Sect. 2 we give some definitions and preliminary results. In Sect. 3 we show that any graph that it is not a non-separable p-connected graph is 2-clique-colourable. In Sect. 4 we show that the class of p-trees is 2-clique-colourable. In Sect. 5 we find the clique-chromatic number and the clique-colouring of the remaining classes of graphs mentioned above. These results lead to polynomial time algorithms for finding the clique-colouring and the clique-chromatic number of the above classes.

Preliminaries

Basic notions

Throughout this paper let G=(V,E) be a finite simple undirected graph and let |V|=n and |E|=m. The complement graph of G=(V,E) is the graph $\overline{G} = (V, \overline{E})$, where $uv \in\overline{E}$ if and only if $uv\not\in E$.

For a vertex v∈V the neighbourhood of v in G is N(v)={u|uv∈E} and for U⊂V, N(U) is the set of vertices in V−U that are adjacent to at least one vertex of U. A clique of G is a set of pairwise adjacent vertices of G and a stable set is a set of pairwise non-adjacent vertices of G. Given a subset U of V, let G[U] stand for the subgraph of G induced by U. Let P _n denote the chordless path on n vertices and n−1 edges. Let C _n denote the chordless cycle with n vertices. If n≥4 then C _n is called a hole and its complement an antihole. A graph is called a complete graph if every pair of distinct vertices is connected by an edge. A graph is called split graph if its vertex set can be partitioned in a clique K and a stable set S. A split graph is a spider if and only if |K|=|S|≥2 and there exists a bijection f between S and K such that for each v∈S, either N(v)={f(v)} (thin legs) or N(v)=K−{f(v)} (thick legs). The simplest spider is a P₄. In a P₄ with vertices u,v,w,x and edges uv,vw,wx, the vertices v and w are called midpoints whereas the vertices u and x are called endpoints.

A module of G is a set of vertices M of V such that each vertex in V−M is either adjacent to all vertices of M, or to none. The whole V and every singleton vertex are trivial modules. Whenever G has only trivial modules it is called a prime graph. A non-trivial module is also called an homogeneous set. We say that M is a strong module if for any other module A the intersection M∩A is empty or equals either M or A. For a non-trivial graph G, the family {M₁,M₂,…,M _p } of all maximal (proper) strong modules is a partition of V(G). This partition is the modular decomposition ofG. We will often identify the modules M _i with the induced subgraphs G _i =G[M _i ].

Whenever a graph G has a non-trivial maximal module M, in order to get some of its structural properties, it is useful to contract the module M to one representative vertex m obtaining a new graph H where V(H)=V(G)−M∪{m} and E(H)=E(G/M)∪{ym|y∈N(M)}. The graph G′ obtained from G by shrinking every maximal non-trivial module to a single vertex is called the characteristic graph of G. If G is a prime graph the G′=G.

If G is a prime graph we consider the substitution operation which consists of substituting any vertex x of G by any graph H in such a way that all the vertices of H have the same adjacencies of x in G. Therefore in the new graph G′ obtained with this substitution operation V(H) is a homogeneous set.

Let G and G′ be two vertex disjoint graphs. We can define the parallel composition of G and G′ as the graph G∪G′ so that V(G∪G′)=V(G)∪V(G′) and E(G∪G′)=E(G)∪E(G′). The serial composition of G and G′ is the graph G+G′ defined by V(G+G′)=V(G)∪V(G′) and E(G+G′)=E(G)∪E(G′)∪{vv′ for each v∈V(G) and v′∈V(G′)}.

p-Connectness and primeval decomposition

Following the terminology of Jamison and Olariu [25], a graph G is p-connected (or, more extensively, P₄-connected) if, for each partition V₁,V₂ of V into two non-empty sets, there exists a chordless path of four vertices P₄ which contains vertices from V₁ and V₂. Such P₄ is a crossing between V₁ and V₂. An equivalent characterization of p-connected graphs can be given in terms of p-chains, a natural analogue of paths in the context of usual connectedness of graphs. A p-chain connecting vertices u and v is a sequence of pairwise different vertices (v₁,v₂,…,v _k ) such that

• 1. u=v₁, v=v _k , and

• 2. X _i ={v _i ,v_i+1,v_i+2,v_i+3} induces a P₄, for i=1,2,…,k−3.

By Babel and Olariu [5, 6] a graph is p-connected if and only if for every pair of vertices in the graph there exists a p-chain connecting them.

The p-connected components of a graph G are the maximal induced p-connected subgraphs. Vertices of G that do not belong to any p-connected component of G are termed weak vertices. A p-connected graph is called separable if its vertex set can be partitioned into two non-empty sets V₁ and V₂ in such a way that each crossing P₄ has its midpoints in V₁ and its endpoints in V₂. For separable p-connected graph the following theorem holds.

Theorem 1

 [25] Ap-connected graph is separable if and only if its characteristic graph is a split graph.

Separable p-connected components play a crucial role in the theory of p-connectedness and their introduction is justified by the following general theorem.

Theorem 2

 [25] For an arbitrary graphGexactly one of the following conditions is satisfied:

• 1. Gis disconnected;

• 2. $\overline{G}$is disconnected;

• 3. There is a unique proper separablep-connected componentHofGwith vertex partition (V₁,V₂) such that every vertex outsideHis adjacent to all vertices inV₁and to no vertex inV₂;

• 4. Gisp-connected.

This theorem implies a decomposition scheme for arbitrary graphs called primeval decomposition.

For disconnected G, the maximal strong modules are the connected components. In this case G=G₁∪G₂∪⋯∪G _p is called parallel.

If $\overline{G}$ is disconnected, the maximal strong modules of G are the connected components of $\overline{G}$. In this case G=G₁+G₂+⋯+G _p is called serial.

If both G and $\overline{G}$ are connected, then either G can be decomposed according to condition 3 of Theorem 2 and in this case G is called a decomposable neighbourhood graph or G is a p-connected graph that may be either separable or non-separable.

By repeating the process we can associate to any non-empty graph G its unique primeval decomposition treeT(G). The root of T(G) is G, the leaves are the p-connected components and the weak vertices of G and the internal nodes of T(G) are labeled with P, S or N (for parallel, serial, or decomposable neighbourhood graph, respectively).

The clique-colouring of graphs which are not non-separable p-connected graphs

In this section we will show that the clique-chromatic number of any graph which is not a non-separable p-connected graph is equal to 2. In the next sections we shall consider classes of graphs that contain non-separable p-connected graphs of special type.

From now on we will assume that the graphs we are considering are connected, otherwise we could always consider separately each connected component.

First we show that in order to find an optimal clique-colouration of a graph, it is enough to have an optimal clique-colouration of its characteristic graph.

Theorem 3

Every graphGhas the same clique-chromatic number of its characteristic graphG′. Given an optimal clique-colouration ofG′, an optimal clique-colouration ofGis obtained assigning to every vertex belonging to a maximal strong module ofGthe same colour of its representative vertex inG′.

Proof

Let G be a graph and let G′ be its characteristic graph. Let {M₁,M₂,…,M _p } be the family of all maximal (proper) strong modules of G and let {v₁,v₂,…,v _p } be the set of the corresponding characteristic vertices in G′. Let us assume that an optimal clique-colouration of G′ is known. Let $K' = \{v_{i_{1}},v_{i_{2}},\ldots, v_{i_{s}}\}$ be any maximal clique of G′. By replacing each vertex ${v_{i_{j}}}$ of every K′ with any maximal clique of the corresponding module ${M_{i_{j}}}$ we obtain all the maximal cliques of G. In fact each set of vertices generated is a clique since each pair of vertices are adjacent either by construction or by definition of module. The maximality follows by the maximality of each clique K′ of G′ and the maximality of the cliques replacing each vertex of K′. Now, we can extend the clique colouration of G′ to G by assigning the colour of each representative vertex of G′ to every vertex of the corresponding homogeneous set in G. No maximal clique of size at least 2 is monocoloured since every maximal clique of G contains an induced subgraph isomorphic to a maximal clique of G′ that is not monocoloured by hypothesis. The optimality of the colouration of G follows from the optimality of the colouration of G′. □

Theorem 4

Every serial graph is 2-clique colourable.

Proof

Let G=G₁+G₂+⋯+G _p , p≥2, be a serial graph. The characteristic graph of G is a complete graph that is 2-clique colourable. Therefore G is also 2-clique-colourable by Theorem 3. In fact, it is sufficient to colour the vertices of G₁ with colour 1 and the remaining vertices with colour 2. □

Lemma 1

Every split graph is 2-clique colourable.

Proof

Let G be a split graph. Then its vertex set can be partitioned into a maximal clique K and an independent set S. Every maximal clique of G is either K or a subset of K with at most one vertex of S. Choose any vertex of K, say x, and colour it with colour 1. Colour the remaining vertices of K with colour 2. Colour the vertices of N _S (x) with colour 2 and the remaining vertices of S with colour 1. We claim that this colouring is a 2-clique-colouring of G. In fact, if K is a maximal clique of G it has two different colours. Any other maximal clique of G contains vertices of K and only one vertex of S. If it contains x then it is 2-clique-colourable since all its adjacent vertices have colour 2. Else, if it contains only vertices of K coloured with colour 2, then the vertex of S by construction has colour 1. So it is 2-clique colourable. □

Theorem 5

Every separablep-connected graph is 2-clique-colourable.

Proof

The characteristic graph of every separable p-connected graph G is a split graph by Theorem 1. Any split graph is 2-clique-colourable by Lemma 1. Therefore G is also 2-clique-colourable by Theorem 3. □

Theorem 6

Every decomposable neighbourhood graph is 2-clique colourable.

Proof

Let G=(V,E) be a decomposable neighbourhood graph. Then by condition 3 of Theorem 2 there is a unique proper separable p-connected component H of G with vertex partition (V₁,V₂) such that every vertex in V₃=V∖(V₁∪V₂) is adjacent to all vertices in V₁ and to no vertex in V₂. Then V₃ is a module in G. The characteristic graph of G is obtained by shrinking V₃ to a single vertex, and by substituting H with its characteristic graph, which is a split graph by Theorem 1. Then the characteristic graph of G is also a split graph. Therefore G is 2-clique colourable by Lemma 1 and Theorem 3. □

The clique-colouring of p-trees and p-forests

The purpose of this section is to show that a special subclass of the P₄-connected graphs, called p-trees is 2-clique-colourable. This class, introduced by Babel in [3], is provided with structural properties that can be expressed in a quite analogous way to the characterization of ordinary trees. A vertex is called a p-end-vertex if it belongs to exactly one P₄. A p-cycle is a p-connected graph with no p-end-vertices, and which is minimal with this property. The class of p-trees is the class of p-connected graphs without induced p-cycles and containing exactly n−3 P₄’s. A p-forest is a graph whose p-connected components are p-trees [3].

In the following we shall use the characterization of p-trees given in [4], based on the structure of the p-chains in a p-tree, which we recall for reader’s convenience.

A p-chain X=(v₁,v₂,…,v _k ) is simple if the only P₄’s contained in G[{v₁,v₂,…,v _k }] are induced by the set of vertices {v _i ,v_i+1,v_i+2,v_i+3} for i=1,2,…,k−3. In other words a p-chain X is simple if and only if the vertices of the p-chain induce precisely k−3 P₄’s.

Let P _k be a path with vertex set V={v₁,v₂,…,v _k } and edge set E={v₁v₂,v₂v₃,…,v_k−1v _k } and let Q _k be a split graph with vertex set V={v₁,v₂,…,v _k }, where A={v_2i−1∈V|1≤2i−1≤k} is a stable set, B={v_2i∈V|2≤2i≤k} is a clique and the edges connecting each vertex of B to the vertices of A are {v_2iv_2i−1 and v_2iv_2j+1,j>i} (see Fig. 1). The ordered sequence (v₁,v₂,…,v _k ) of the vertices of P _k (k≥4), Q _k (k≥5) and their complements are simple p-chains. It has been proved that graphs whose vertices can be ordered in a simple p-chains are p-trees and it turns out that every p-tree can be generated starting from a simple p-chain extended by a number of p-end-vertices which can eventually be replaced by cographs [4].

Fig. 1

The graph Q₉

A spikedp-chainP _k is a path P _k , k≥5, eventually extended, by introducing two additional vertices x and y such that x is adjacent to v₂ and v₃ and y is adjacent to v_k−1 and v_k−2; moreover we request that x and y do not belong to a common P₄ (see Fig. 2).

Fig. 2

The spiked p-chain P₈

A spikedp-chainQ _k is a split graph Q _k , k≥6, with eventually, additional vertices z₂,z₃,…,z_k−5 such that

A spiked p-chain is shown in Fig. 3. A spiked p-chain $\overline{P}_{k}$ (or $\overline{Q}_{k}$) is the complement of a spiked p-chain P _k (or Q _k ).

Fig. 3

The spiked p-chain Q₉

It is easy to verify that v₁,x,y,v _k and v₁,z₂,z₃,…,z_k−5,v _k are the only p-end-vertices of spiked P _k ($\overline{P}_{k}$) and Q _k ($\overline{Q}_{k}$) respectively.

Finally we have the following characterization of p-trees given by Babel.

Theorem 7

[4] A graph is ap-tree if and only if it is either aP₄with one vertex replaced by a cograph or a spikedp-chain with thep-end-vertices replaced by cographs.

Theorem 8

Everyp-tree is 2-clique-colourable.

Proof

From Theorem 7, it follows that the only homogeneous sets of a p-tree are the cographs eventually replacing the p-end-vertices of G. By Theorem 3 graph G has the same clique-chromatic number of its characteristic graph. Therefore it is enough to consider the clique-colouration of the spiked p-chains P _k , Q _k and their complements.

Notice that a prime p-tree is separable if and only if it is either a spiked p-chain Q _k or $\overline{Q}_{k}$, (k≥6) or a P₄. If G is a separable p-tree then it is 2-clique-colourable by Theorem 5. If G is not separable then G is a spiked p-chains P _k or $\overline{P}_{k}$, (k≥5). Let now consider the case when G is a spiked p-chains P _k . If the p-end-vertices x and y are not present then G is a path P _k that can be 2-clique-coloured by assigning to the vertices of the path alternatively colours 1 and 2. If the p-end-vertices x or y are present they belong to a maximal clique of G that already contains a 2-coloured edge so they can be coloured with colour 1 or 2. Let G be a spiked p-chain $\overline{P}_{k}$. If the p-end-vertices x and y are not present then G is a $\overline{P}_{k}$. Let v₁,v₂,…,v _k be the sequence of the vertices of P _k . The vertex set of a maximal clique of $\overline{P}_{k}$ is any maximal ordered sequence of vertices $S= v_{i_{1}},v_{i_{2}},\ldots, v_{i_{s}}$ such that for each two consecutive vertices $v_{i_{p}}$ and $v_{i_{q}}$, i _q −i _p is either 2 or 3. Since S is a maximal sequence then $v_{i_{1}}$ must belong to the set {v₁,v₂} and $v_{i_{s}}$ to the set {v_k−1,v _k }. Then every maximal clique of $\overline{P}_{k}$ contains one of the edges v₁v _k or v₁v_k−1 or v₂v _k or v₂v_k−1. Therefore, in order to obtain a 2-clique colouration of $\overline{P}_{k}$ it is enough to colour the vertices v₁ and v₂ with colour 1, the vertices v _k and v_k−1 with colour 2 and any other vertex with colour 1 or 2. If the p-end-vertices x or y are present they belong to a maximal clique of G that already contains a 2-coloured edge so they can be coloured with colour 1 or 2. □

Theorem 9

Everyp-forest is 2-clique-colourable.

Proof

A p-forest G is either a serial graph or a decomposable neighbourhood graph or a p-tree. Then G is 2-clique-colourable by Theorems 4, 6 and 8 respectively. □

The clique-colouring of graphs with few P₄’s

In this section we shall consider graphs which, in a local sense, contain only a restricted number of P₄’s. It all started with the class of cographs, which is a class of graphs where no P₄ is allowed to exist. These graphs have been investigated in [11, 12] and many nice structural results are known. In particular it has been shown in [11] that every connected cograph is a serial graph and therefore it is 2-clique-colourable by Theorem 4.

The study of cographs has been extended by many authors. Hoáng [20] introduced the class of P₄-sparse graphs, which is the class such that no set of five vertices induces more than one P₄. The non-trivial leaves of its associated primeval tree are spiders [24]. Jamison and Olariu [21–23] introduced the class of P₄-reducible graphs, P₄-extendible and P₄-lite. The P₄-reducible graphs are the graphs such that no vertex belongs to more than one P₄, and its p-connected components are P₄’s. The P₄-extendible are graphs where each p-connected component consists of at most five vertices. Each p-connected component is either P₅ or $\overline{P_{5}}$ or C₅, or P₄ with one vertex eventually substituted by a homogeneous set with cardinality two. The P₄-lite are graphs such that every induced subgraph with at most six vertices either contains at most two P₄’s or is isomorphic to a spider. The p-connected components of a P₄-lite graph are either a spider (possibly with one vertex replaced by a homogeneous set of cardinality 2) or one of the graphs P₅, $\overline{P_{5}}$. The P₄-laden [16] are graphs such that every induced subgraph with at most six vertices either contains at most two P₄’s or it is isomorphic to a split graph. The p-connected components of a P₄-laden graph are spiders (possibly with one vertex replaced by a homogeneous set of cardinality 2), split graphs, P₅’s or $\overline{P_{5}}$’s [16]. The above classes are ordered as follows:

$$\mathrm{cographs} \subset P_4\mbox{-reducible} \subset P_4\mbox{-sparse} \subset P_4\mbox{-lite} \subset P_4\mbox{-laden}.$$

Another generalization of the above classes are the extended P₄-reducible  [17], extended P₄-sparse graphs [17], P₄-tidy graph [18] and the extended P₄-laden graphs [16]. These classes are obtained from P₄-reducible and P₄-sparse graphs, P₄-lite and P₄-laden respectively, by also allowing C₅’s as p-connected components. All the previously mentioned classes are included in the class of extended P₄-laden graphs.

Using the primeval decomposition we have the following.

Theorem 10

Every extended-P₄-laden graphGis 2-clique-colourable with the exception ofC₅which is 3-clique-colourable.

Proof

If G is a serial graph or a decomposable neighbourhood graph, it is 2-clique-colourable by Theorem 4 and Theorem 6 respectively. If G is a p-connected graph different from C₅, then either it is isomorphic to a P₅ or $\overline{P_{5}}$ that are trivially 2-clique-colourable or it is a separable p-connected graph isomorphic either to a spider graph (possibly with one vertex replaced by a homogeneous set of cardinality 2), or to a split graph. The characteristic graph in both cases is a split graph and therefore it is 2-clique-colourable by Lemma 1. If G is a C₅ then G is 3-clique-colourable. □

Corollary 1

Every cograph, P₄-reducible, P₄-sparse andP₄-lite graphs are 2-clique-colourable.

Corollary 2

EveryP₄-extendible, extendedP₄-reducible and extendedP₄-sparse graphs are 2-clique-colourable with the exception ofC₅which is 3-clique-colourable.

A generalization of some of the above classes was given by Babel and Olariu. They proposed to call a graph a (q,t)-graph if no set of at most q vertices induces more than t distinct P₄’s. In this terminology, the cographs are precisely the (4,0)-graphs, the P₄-sparse graphs coincide with the (5,1)-graphs. In particular the (7,4)-graphs properly contain all cographs, P₄-reducible, P₄-sparse, p-trees and p-forests. The (9,6)-graphs additionally contain all P₄-extendible, extended P₄-reducible and extended P₄-sparse graphs.

The p-connected (q,q−3)-graph has been characterized as follows.

Theorem 11

[4] LetGbe ap-connected graph withnvertices.

• (i) Ifn≥7, thenGis a (n,n−3) graph if and only ifGis ap-tree.

• (ii) Ifn>q, q∈{7,9}, thenGis a (q,q−3)-graph if and only if precisely one of the following conditions holds

  • (a) Gis ap-tree;

  • (b) Gis a hole or antihole;

  • (c) Gis a spider.

• (iii) Ifn>q, q=8, orq≥10, thenGis a (q,q−3)-graph if and only if precisely one of the following conditions holds

  • (a) Gis ap-tree;

  • (b) Gis a hole or antihole.

In order to find the clique-colouration of (q,q−3)-graphs we will need the following result.

Theorem 12

Every antihole$\overline{C}_{k}$, withk>5 is 2-clique-colourable.

Proof

Let v₁,v₂,…,v _k , k≥6, be the sequence of the vertices of C _k . Any 2-colouring where v₁,v₂,v₃ have colour 1 and v₄,v₅,v₆ have colour 2 is a 2-clique-colouring. In fact a clique that misses 3 consecutive vertices v _i ,v_i+1,v_i+2 is not maximal because we could include v_i+1 obtaining a bigger clique. Hence, any clique has (at least) one vertex in {v₁,v₂,v₃} and (at least) one vertex in {v₄,v₅,v₆} and it is 2-clique-coloured. □

Note that Theorem 12 can be obtained also as a particular case of results contained in [1, 7].

Now we are ready to prove the following theorem.

Theorem 13

Letq≥7 be a fixed integer andGa (q,q−3)-graph withnvertices, n≥q. Gis 2-clique-colourable, unless it is aC _n withnodd (which is 3-clique-colourable).

Proof

If n=q then G is a p-tree, which is 2-clique-colourable by Theorem 8. If n>q then G is either a serial graph or a decomposable neighbourhood graph or a p-connected graph. In the first two cases it is 2-clique-colourable by Theorems 4 and 6 respectively. In the last case, G has more than 7 vertices and it is either a p-tree, or a hole, or an antihole, or a spider by Theorem 11. Every p-tree is 2-clique-colourable by Theorem 8. Every spider is a split graph and therefore it is 2-clique-colourable by Lemma 1. Every antihole is 2-clique-colourable by Theorem 12. Finally every hole is 2-clique-colourable if it is even, otherwise is 3-clique-colourable. □

Final remarks

We would like to point out that, in order to study the clique-colouration of a graph, it is enough to consider the clique-colouration of its characteristic graph that can be obtained in linear time through the modular decomposition of that graph [13, 28]. We have showed that every serial graph, every separable p-connected graph and every neighbourhood decomposable graph is 2-clique-colourable. Furthermore we have solved the clique-colouring problem for classes of graphs that contain very special non-separable p-connected graphs. Moreover, if the clique-colouration of a prime graph G is known, then any graph obtained by substituting any vertex of G with any graph H has the same clique-colouration of G. The problem remains open for general non-separable p-connected graphs.