List-k-Coloring H-Free Graphs for All k > 4

Abstract

Given an integer k > 4 and a graph H, we prove that, assuming P =/ NP, the LIST-k-COLORING PROBLEM restricted to H-free graphs can be solved in polynomial time if and only if either every component of H is a path on at most three vertices, or removing the isolated vertices of H leaves an induced subgraph of the five-vertex path. In fact, the "if" implication holds for all k>_ 1.