Colouring graphs with prescribed induced cycle lengths

Abstract

In this paper we study the chromatic number of graphs with two prescribed induced cycle lengths. It is due to Sumner that trianglefree and P5-free or triangle-free, P6-free and C6-free graphs are 3colourable. A canonical extension of these graph classes is G I (4;5), the class of all graphs whose induced cycle lengths are 4 or 5. Our main result states that all graphs of G I (4;5) are 3-colourable. Moreover, we present polynomial time algorithms to 3-colour all triangle-free graphs G of this kind, i.e., we have polynomial time algorithms to 3-colour every G 2 G I (n1;n2) with n1;n2 ‚ 4 (see Table 1). Furthermore, we consider the related problem of finding a ´-binding function for the class G I (n1;n2). Here we obtain the surprising result that there exists no linear ´-binding function for G I (3;4).