Transitive orientations in bull-reducible Berge graphs

Abstract

A bull is a graph with five vertices r, y, x, z, s and five edges ry, yx, yz, xz, zs. A graph G is bull-reducible if no vertex of G lies in two bulls. We prove that every bull-reducible Berge graph G that contains no antihole is weakly chordal, or has a homogeneous set, or is transitively orientable. This yields a fast polynomial time algorithm to color exactly the vertices of such a graph.