Graph colourings with forbidden k-coloured subgraphs

Abstract

For a set F of graphs and a natural number k, an (F, k)-colouring of a graph G is a proper colouring of V (G) such that no subgraph of G isomorphic to an element of F is coloured with at most k colours. Equivalently, if P is the class of all graphs that do not contain an element of F as a subgraph, a χP,k colouring of G is a proper colouring such that the union of at most k colour classes induces a graph in P. The smallest number of colours in such a colouring of G, if it exists, is denoted by χP,k (G). We give some general results on χP,k-colourings and investigate values of χP,k (G) for some choices of P and classes of graphs G.