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
,vi+1,vi+2 is not maximal because we could include vi+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. □