List coloring in the absence of a linear forest.

The k-Coloring problem is to decide whether a graph can be colored with at most k colors such that no two adjacent vertices receive the same color. The List k -Coloring problem requires in addition that every vertex u must receive a color from some given set L(u) ⊆ {1,…,k}. Let P n denote the path on n vertices, and G + H and rH the disjoint union of two graphs G and H and r copies of H, respectively. For any two fixed integers k and r, we show that List k -Coloring can be solved in polynomial time for graphs with no induced rP 1 + P 5, hereby extending the result of Hoàng, Kamiński, Lozin, Sawada and Shu for graphs with no induced P 5. Our result is tight; we prove that for any graph H that is a supergraph of P 1 + P 5 with at least 5 edges, already List 5-Coloring is NP-complete for graphs with no induced H. We also show that List k -Coloring is fixed parameter tractable in k + r on graphs with no induced rP 1 + P 2, and that k-Coloring restricted to such graphs allows a polynomial kernel when parameterized by k. Finally, we show that List k -Coloring is fixed parameter tractable in k for graphs with no induced P 1 + P 3