Given an integer \(k>4\) and a graph H, we prove that, assuming P \(\ne \) 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 \(k\ge 1\) .
