On the Complexity of the General Coloring Problem

A graph H is called an interpretation of a graph G if a morphic image of H is (isomorphic to) a subgraph of G. A graph H can be seen to be n-colorable iff H is an interpretation of K, (the complete graph on n vertices): hence colorability is a special case of the concept of interpretation. In this paper the complexity of the general coloring problem, i.e., of deciding for a given graph G whether some graph H is an interpretation of G, is investigated. It is shown that for many very simple undirected graphs G this question is NP-complete (this was previously known for the graphs K, only). In fact, it seems to be NP-complete for all but trivial exceptions. In the directed case, non-trivial graphs G for which the problem is in P, and rather simple graphs G for which the problem is NP-complete are presented. A characterization of all graphs G for which the problem at issue is in P remains an evasive open problem.