Three-colourability and forbidden subgraphs. II: polynomial algorithms Original Research Article

In this paper we study the chromatic number for graphs with forbidden induced subgraphs. We focus our interest on graph classes (defined in terms of forbidden induced subgraphs) for which the question of 3-colourability can be decided in polynomial time and, if so, a proper 3-colouring can be determined also in polynomial time. Note that the 3-colourability decision problem is a well-known NP-complete problem, even for special graph classes, e.g. triangle-free and K1,5-free (Discrete Math. 162 (1–3) (1996) 313). Therefore, it is unlikely that there exists a polynomial algorithm deciding whether there exists a 3-colouring of a given graph in general. We present three different approaches to reach our goal. The first approach is purely a structural analysis of the graph class in consideration; the second one is a structural analysis of only the non-perfect K4-free members of the considered graph class; finally the last approach is based on propositional logic and bounded dominating subgraphs.