List-k-Coloring H-Free Graphs for All k>4 - Chudnovsky, Maria; Hajebi, Sepehr; Spirkl, Sophie (Combinatorica (Budapest. 1981), 2024)
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 .