The complexity of restricted graph homomorphisms Original Research Article

A well-known result of Hell and Nešetřil 1992 states that if H is a fixed non-bipartite graph, then it is an NP-complete problem to decide if there exists a homomorphism of a given graph G to H. The problem is polynomial if H is bipartite. The same authors have conjectured that, if H is non-bipartite, the problem remains NP-complete even if restricted to instances where the input graph G admits a homomorphism to a fixed graph Y, unless there is a homomorphism of Y to H. We describe some progress on this conjecture, and the possibility of extending it to digraphs.