On the complexity of colouring by superdigraphs of bipartite graphs

Abstract Let H be a directed graph whose vertices are called colours. An H-colouring of a digraph G is an assignment of these colours to the vertices of G so that if g is adjacent to g ′ in G then colour( g ) is adjacent to colour( g ′) in H (i.e., a homomorphism G → H ). In this paper we continue the study of the H-colouring problem , that is, the decision problem ‘Is there an H -colouring of a given digraph G ?’ It follows from a result of Hell and Nesetřil that this problem is NP-complete whenever H contains a symmetric odd cycle. We consider digraphs for which the symmetric part of H is bipartite, that is, digraphs H which can be constructed from the equivalence digraph of an undirected bipartite graph by adding some arcs. We establish some sufficient conditions for these H -colouring problems to be NP-complete. A complete classification is established for the subclass of ‘partitionable digraphs’, which we introduce.