3-Colourability and Forbidden Subgraphs

In this paper we investigate the chromatic aspects for graphs with forbidden induced subgraphs with emphasis on the question of 3-colourability. In the main part all possible pairs (A, B) of forbidden induced subgraphs, s.t. every A- and B -free graph is 3-colourable, are determined. Moreover, this is embedded in a more general problem and the main result extends Vizingʹs classical edge colouring result. Secondly, we give an example for a polynomial algorithm concerning a special graph class deciding, whether there exists a 3-colouring. Finally, graph classes GI(n1,n2) containing graphs whose induced cycle lengths are n1 or n2, are treated. Surprisingly, all triangle-free graphs of this kind are 3-colourable.