# On clique-colouring of graphs with few *P*_{4}’s

- Sulamita Klein
^{1}Email author and - Aurora Morgana
^{2}

**18**:53

https://doi.org/10.1007/s13173-011-0053-3

© The Brazilian Computer Society 2011

**Received: **1 December 2011

**Accepted: **8 December 2011

**Published: **22 December 2011

## 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 *P*_{4}’s. In particular we shall consider the classes of extended *P*_{4}-laden graphs, *p*-trees (graphs which contain exactly *n*−3 *P*_{4}’s) and (*q*,*q*−3)-graphs, *q*≥7, such that no set of at most *q* vertices induces more that *q*−3 distincts *P*_{4}’s. As corollary we shall derive the clique-chromatic number and the clique-colouring of the classes of cographs, *P*_{4}-reducible graphs, *P*_{4}-sparse graphs, extended *P*_{4}-reducible graphs, extended *P*_{4}-sparse graphs, *P*_{4}-extendible graphs, *P*_{4}-lite graphs, *P*_{4}-tidy graphs and *P*_{4}-laden graphs that are included in the class of extended *P*_{4}-laden graphs.

### Keywords

Graphs Clique-colouring Primeval decomposition*P*

_{4}-structure

## 1 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 *P*_{4}’s. In particular we shall consider the classes of extended *P*_{4}-laden [16], *p*-trees [3] (graphs which contain exactly *n*−3 *P*_{4}’s) and (*q*,*q*−3)-graphs, *q*≥7, [2] such that no set of at most *q* vertices induces more that *q*−3 distincts *P*_{4}’s. As corollary we shall derive the clique-chromatic number and the clique-colouring of the classes of cograph [11, 12], *P*_{4}-reducible graphs [22], *P*_{4}-sparse graphs [20, 24], extended *P*_{4}-reducible graphs [17], extended *P*_{4}-sparse graphs [17], *P*_{4}-extendible graphs [23], *P*_{4}-lite graphs [21], *P*_{4}-tidy graphs [18] and *P*_{4}-laden graphs [16] that are included in the class of extended *P*_{4}-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.

## 2 Preliminaries

### 2.1 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 *P*_{4}. In a *P*_{4} 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 {*M*_{1},*M*_{2},…,*M*_{
p
}} of all maximal (proper) strong modules is a partition of *V*(*G*). This partition is the *modular decomposition of**G*. 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*′)}.

### 2.2 *p*-Connectness and primeval decomposition

*G*is

*p-connected*(or, more extensively,

*P*

_{4}

*-connected*) if, for each partition

*V*

_{1},

*V*

_{2}of

*V*into two non-empty sets, there exists a chordless path of four vertices

*P*

_{4}which contains vertices from

*V*

_{1}and

*V*

_{2}. Such

*P*

_{4}is a

*crossing*between

*V*

_{1}and

*V*

_{2}. 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 (

*v*

_{1},

*v*

_{2},…,

*v*

_{ k }) such that

- 1.
*u*=*v*_{1},*v*=*v*_{ k }, and - 2.
*X*_{ i }={*v*_{ i },*v*_{i+1},*v*_{i+2},*v*_{i+3}} induces a*P*_{4}, for*i*=1,2,…,*k*−3.

*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 *V*_{1} and *V*_{2} in such a way that each crossing *P*_{4} has its midpoints in *V*_{1} and its endpoints in *V*_{2}. For separable *p*-connected graph the following theorem holds.

### Theorem 1

[25] *A**p*-*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

*For an arbitrary graph*

*G*

*exactly one of the following conditions is satisfied*:

- 1.
*G**is disconnected*; - 2.
\(\overline{G}\)

*is disconnected*; - 3.
*There is a unique proper separable**p*-*connected component**H**of**G**with vertex partition*(*V*_{1},*V*_{2})*such that every vertex outside**H**is adjacent to all vertices in**V*_{1}*and to no vertex in**V*_{2}; - 4.
*G**is**p*-*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*=*G*_{1}∪*G*_{2}∪⋯∪*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*=*G*_{1}+*G*_{2}+⋯+*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 tree**T*(*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).

## 3 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 graph**G**has the same clique*-*chromatic number of its characteristic graph**G*′. *Given an optimal clique*-*colouration of**G*′, *an optimal clique*-*colouration of**G**is obtained assigning to every vertex belonging to a maximal strong module of**G**the same colour of its representative vertex in**G*′.

### Proof

Let *G* be a graph and let *G*′ be its characteristic graph. Let {*M*_{1},*M*_{2},…,*M*_{
p
}} be the family of all maximal (proper) strong modules of *G* and let {*v*_{1},*v*_{2},…,*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*=*G*_{1}+*G*_{2}+⋯+*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 *G*_{1} 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 separable**p*-*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 (*V*_{1},*V*_{2}) such that every vertex in *V*_{3}=*V*∖(*V*_{1}∪*V*_{2}) is adjacent to all vertices in *V*_{1} and to no vertex in *V*_{2}. Then *V*_{3} is a module in *G*. The characteristic graph of *G* is obtained by shrinking *V*_{3} 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. □

## 4 The clique-colouring of *p*-trees and *p*-forests

The purpose of this section is to show that a special subclass of the *P*_{4}-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 *P*_{4}. 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 *P*_{4}’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*=(*v*_{1},*v*_{2},…,*v*_{
k
}) is *simple* if the only *P*_{4}’s contained in *G*[{*v*_{1},*v*_{2},…,*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 *P*_{4}’s.

*P*

_{ k }be a path with vertex set

*V*={

*v*

_{1},

*v*

_{2},…,

*v*

_{ k }} and edge set

*E*={

*v*

_{1}

*v*

_{2},

*v*

_{2}

*v*

_{3},…,

*v*

_{k−1}

*v*

_{ k }} and let

*Q*

_{ k }be a split graph with vertex set

*V*={

*v*

_{1},

*v*

_{2},…,

*v*

_{ k }}, where

*A*={

*v*

_{2i−1}∈

*V*|1≤2

*i*−1≤

*k*} is a stable set,

*B*={

*v*

_{2i}∈

*V*|2≤2

*i*≤

*k*} is a clique and the edges connecting each vertex of

*B*to the vertices of

*A*are {

*v*

_{2i}

*v*

_{2i−1}and

*v*

_{2i}

*v*

_{2j+1},

*j*>

*i*} (see Fig. 1). The ordered sequence (

*v*

_{1},

*v*

_{2},…,

*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].

*spiked*

*p-chain*

*P*

_{ k }is a path

*P*

_{ k },

*k*≥5, eventually extended, by introducing two additional vertices

*x*and

*y*such that

*x*is adjacent to

*v*

_{2}and

*v*

_{3}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

*P*

_{4}(see Fig. 2).

*spiked*

*p-chain*

*Q*

_{ k }is a split graph

*Q*

_{ k },

*k*≥6, with eventually, additional vertices

*z*

_{2},

*z*

_{3},…,

*z*

_{k−5}such that

*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 }).

It is easy to verify that *v*_{1},*x*,*y*,*v*_{
k
} and *v*_{1},*z*_{2},*z*_{3},…,*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 a**p*-*tree if and only if it is either a**P*_{4}*with one vertex replaced by a cograph or a spiked**p*-*chain with the**p*-*end*-*vertices replaced by cographs*.

### Theorem 8

*Every**p*-*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 *P*_{4}. 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 *v*_{1},*v*_{2},…,*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 {*v*_{1},*v*_{2}} 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 *v*_{1}*v*_{
k
} or *v*_{1}*v*_{k−1} or *v*_{2}*v*_{
k
} or *v*_{2}*v*_{k−1}. Therefore, in order to obtain a 2-clique colouration of \(\overline{P}_{k}\) it is enough to colour the vertices *v*_{1} and *v*_{2} 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

*Every**p*-*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. □

## 5 The clique-colouring of graphs with few *P*_{4}’s

In this section we shall consider graphs which, in a local sense, contain only a restricted number of *P*_{4}’s. It all started with the class of *cographs*, which is a class of graphs where no *P*_{4} 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.

*P*

_{4}-sparse graphs, which is the class such that no set of five vertices induces more than one

*P*

_{4}. The non-trivial leaves of its associated primeval tree are spiders [24]. Jamison and Olariu [21–23] introduced the class of

*P*

_{4}-reducible graphs,

*P*

_{4}-extendible and

*P*

_{4}-lite. The

*P*

_{4}-reducible graphs are the graphs such that no vertex belongs to more than one

*P*

_{4}, and its

*p*-connected components are

*P*

_{4}’s. The

*P*

_{4}-extendible are graphs where each

*p*-connected component consists of at most five vertices. Each

*p*-connected component is either

*P*

_{5}or \(\overline{P_{5}}\) or

*C*

_{5}, or

*P*

_{4}with one vertex eventually substituted by a homogeneous set with cardinality two. The

*P*

_{4}-lite are graphs such that every induced subgraph with at most six vertices either contains at most two

*P*

_{4}’s or is isomorphic to a spider. The

*p*-connected components of a

*P*

_{4}-lite graph are either a spider (possibly with one vertex replaced by a homogeneous set of cardinality 2) or one of the graphs

*P*

_{5}, \(\overline{P_{5}}\). The

*P*

_{4}-laden [16] are graphs such that every induced subgraph with at most six vertices either contains at most two

*P*

_{4}’s or it is isomorphic to a split graph. The

*p*-connected components of a

*P*

_{4}-laden graph are spiders (possibly with one vertex replaced by a homogeneous set of cardinality 2), split graphs,

*P*

_{5}’s or \(\overline{P_{5}}\)’s [16]. The above classes are ordered as follows:

Another generalization of the above classes are the extended *P*_{4}-reducible [17], extended *P*_{4}-sparse graphs [17], *P*_{4}-tidy graph [18] and the extended *P*_{4}-laden graphs [16]. These classes are obtained from *P*_{4}-reducible and *P*_{4}-sparse graphs, *P*_{4}-lite and *P*_{4}-laden respectively, by also allowing *C*_{5}’s as *p*-connected components. All the previously mentioned classes are included in the class of extended *P*_{4}-laden graphs.

Using the primeval decomposition we have the following.

### Theorem 10

*Every extended*-*P*_{4}-*laden graph**G**is* 2-*clique*-*colourable with the exception of**C*_{5}*which 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 *C*_{5}, then either it is isomorphic to a *P*_{5} 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 *C*_{5} then *G* is 3-clique-colourable. □

### Corollary 1

*Every cograph*, *P*_{4}-*reducible*, *P*_{4}-*sparse and**P*_{4}-*lite graphs are* 2-*clique*-*colourable*.

### Corollary 2

*Every**P*_{4}-*extendible*, *extended**P*_{4}-*reducible and extended**P*_{4}-*sparse graphs are* 2-*clique*-*colourable with the exception of**C*_{5}*which 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 *P*_{4}’s. In this terminology, the cographs are precisely the (4,0)-graphs, the *P*_{4}-sparse graphs coincide with the (5,1)-graphs. In particular the (7,4)-graphs properly contain all cographs, *P*_{4}-reducible, *P*_{4}-sparse, *p*-trees and *p*-forests. The (9,6)-graphs additionally contain all *P*_{4}-extendible, extended *P*_{4}-reducible and extended *P*_{4}-sparse graphs.

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

### Theorem 11

*Let*

*G*

*be a*

*p*-

*connected graph with*

*n*

*vertices*.

- (i)
*If**n*≥7,*then**G**is a*(*n*,*n*−3)*graph if and only if**G**is a**p*-*tree*. - (ii)
*If**n*>*q*,*q*∈{7,9},*then**G**is a*(*q*,*q*−3)-*graph if and only if precisely one of the following conditions holds*- (a)
*G**is a**p*-*tree*; - (b)
*G**is a hole or antihole*; - (c)
*G**is a spider*.

- (a)
- (iii)
*If**n*>*q*,*q*=8,*or**q*≥10,*then**G**is a*(*q*,*q*−3)-*graph if and only if precisely one of the following conditions holds*- (a)
*G**is a**p*-*tree*; - (b)
*G**is a hole or antihole*.

- (a)

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

### Theorem 12

*Every antihole*\(\overline{C}_{k}\), *with**k*>5 *is* 2-*clique*-*colourable*.

### Proof

Let *v*_{1},*v*_{2},…,*v*_{
k
}, *k*≥6, be the sequence of the vertices of *C*_{
k
}. Any 2-colouring where *v*_{1},*v*_{2},*v*_{3} have colour 1 and *v*_{4},*v*_{5},*v*_{6} 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 {*v*_{1},*v*_{2},*v*_{3}} and (at least) one vertex in {*v*_{4},*v*_{5},*v*_{6}} 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

*Let**q*≥7 *be a fixed integer and**G**a* (*q*,*q*−3)-*graph with**n**vertices*, *n*≥*q*. *G**is* 2-*clique*-*colourable*, *unless it is a**C*_{
n
}*with**n**odd* (*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. □

## 6 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.

## Declarations

### Acknowledgements

The authors thank Sylvain Gravier for having introduced the clique colouration subject to the authors and Célia Picinin de Mello for valuable conversations concerning this problem, which helped to improve the presentation of the ideas contained in this work.

This research was partially supported by CNPq and FAPERJ.

## Authors’ Affiliations

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