On the complexity of H-coloring

Abstract

Let H be a fixed graph, whose vertices are referred to as ‘colors’. An H-coloring of a graph G is an assignment of ‘colors’ to the vertices of G such that adjacent vertices of G obtain adjacent ‘colors’. (An H-coloring of G is just a homomorphism G → H). The following H-coloring problem has been the object of recent interest: Instance: A graph G. Question: Is it possible to H-color the graph G? H-colorings generalize traditional graph colorings, and are of interest in the study of grammar interpretations. Several authors have studied the complexity of the H-coloring problem for various (families of) fixed graphs H. Since there is an easy H-colorability test when H is bipartite, and since all other examples of the H-colorability problem that were treated (complete graphs, odd cycles, complements of odd cycles, Kneser graphs, etc.) turned out to be NP-complete, the natural conjecture, formulated in several sources (including David Johnson's NP-completeness column), asserts that the H-coloring problem is NP-complete for any non-bipartite graph H. We give a proof of this conjecture.