Complexity Dichotomy for List-5-Coloring with a Forbidden Induced Subgraph

Abstract

For a positive integer r and graphs G and H, we denote by G+H the disjoint union of G and H and by rH the union of r mutually disjoint copies of H. Also, we say G is H-free if H is not isomorphic to an induced subgraph of G. We use Pt to denote the path on t vertices. For a fixed positive integer k, the List-k-Coloring Problem is to decide, given a graph G and a list L(v)⊆{1,…,k} of colors assigned to each vertex v of G, whether G admits a proper coloring ϕ with ϕ(v)∈L(v) for every vertex v of G, and the k-Coloring Problem is the List-k-Coloring Problem restricted to instances with L(v)={1,…,k} for every vertex v of G. We prove that, for every positive integer r, the List-5-Coloring Problem restricted to rP3-free graphs can be solved in polynomial time. Together with known results, this gives a complete dichotomy for the complexity of the List-5-Coloring Problem restricted to H-free graphs: For every graph H, assuming P≠NP, the List-5-Coloring Problem restricted to H-free graphs can be solved in polynomial time if and only if, H is an induced subgraph of either rP3 or P5+rP1 for some positive integer r. As a hardness counterpart, we also show that the k-Coloring Problem restricted to rP4-free graphs is NP-complete for all k≥5 and r≥2.