The Existence Problem for Graph Homomorphisms

Abstract

Graph homomorphisms have been investigated since the nineteen sixties [l, 2, 4, 7–10, 12–14, 16–20], and have enjoyed renewed interest lately, because of their relation to grammar interpretations [16]. (A different notion of graph homomorphism was investigated by Dirac, Wagner, and others, [3, 21].) A homomorphism G → H is a mapping of the vertex set of G to the vertex set of H such that adjacent vertices have adjacent images. Because a homomorphism c: G → K n is just an n-coloring of G , a homomorphism G + H is also called an H-coloring of G. 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 ? Several authors have studied the complexity of the H -coloring problem for various (families of) fixed graphs H [l, 2, 17, 18]. 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, etc., [l, 17, 18]) turned out to be NP -complete, the natural conjecture, formulated in several sources [17, 18] (including David Johnson's NP -completeness column [12]) asserts that the H -coloring problem is NP -complete for any non-bipartite graph H. We have proved this conjecture, and will publish a full proof elsewhere [ll]. Here, we outline the ideas behind our proof.