(H,C,K) -Coloring: 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+c) steps where c 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.

Research supported by the EU project ALCOM-FT (IST-99-14186) and by the Spanish CYCIT TIC-2000-1970-CE. The research of the 3rd author was supported by the Ministry of Education and Culture of Spain, Grant number MEC-DGES SB98 0K148809.