Locally bounded hereditary subclasses of k-colourable graphs

Abstract

A vertex colouring C 1, C 2,…, C k of a graph G is called l- bounded ( l⩾0) if | C i⧹N(u)|⩽l for all i=1,2,…, k and for every vertex u∈VG⧹ C i ; here N( u) is the neighbourhood of u. Let C( k, l) be the class of all graphs having an l-bounded k-colouring. We show that for all k and l the class C( k, l) can be described in terms of forbidden induced subgraphs. This result implies the existence of polynomial time algorithms recognizing C( k, l). We also find the minimal set of forbidden induced subgraphs for the class C(3,1).