(H,C,K)-colorings: fast, easy, and hard cases

Abstract

We define a variant of the H-coloring problem by fixing the number of preimages of a subset C of the vertices of H, thus allowing parameterization. We provide sufficient conditions to guarantee that the problem can be solved in O(kn+f(k,H)) steps where f is a function depending only on the number k of fixed preimages and the graph H, and in O(n^{k+alpha}) steps where alpha is a constant independent of k. Finally, we prove that whenever the non parameterized vertices induce in G a graph that is bipartite and loopless the problem is NP-complete.