The effect of two cycles on the complexity of colourings by directed graphs

Abstract

Let H be a fixed directed graph. An H-colouring of a directed graph D is a mapping f: V( D)→ V( H) such that f( x) f( y) is an edge of H whenever xy is an edge of D. We study the following H-colouring problem: Instance: A directed graph D. Question: Does there exist an H-colouring of D? In an earlier paper [2] it is shown that among semicomplete digraphs H, it is the existence of two directed cycles in H which makes the H-colouring problem (NP-) hard. In this paper we provide further classes of digraphs in which two directed cycles in H make the H-colouring problem NP-hard. These include both classes of dense and of sparse digraphs. There still appears to be no natural conjecture as to what digraphs H give NP-hard H-colouring problems; however, in view of our results, we are led to make such a conjecture for digraphs without sources and sinks.