List 3-Coloring Graphs with No Induced $P_6+rP_3$

Abstract

For an integer t, we let $P_t$ denote the t-vertex path. We write $H+G$ for the disjoint union of two graphs H and G, and for an integer r and a graph H, we write rH for the disjoint union of r copies of H. We say that a graph G is H-free if no induced subgraph of G is isomorphic to the graph H. In this paper, we study the complexity of k-coloring, for a fixed integer k, when restricted to the class of H-free graphs with a fixed graph H. We provide a polynomial-time algorithm to test if, for fixed r, a $(P_6+rP_3)$-free is three-colorable, and find a coloring if one exists. We also solve the list version of this problem, where each vertex is assigned a list of possible colors, which is a subset of $\{1,2,3\}$. This generalizes results of Broersma, Golovach, Paulusma, and Song, and results of Klimošová, Malik, Masařík, Novotná, Paulusma, and Slívová. Our proof uses a result of Ding, Seymour, and Winkler relating matchings and hitting sets in hypergraphs. We also prove that the problem of deciding if a $(P_5+P_2)$-free graph has a k-coloring is NP-hard for every fixed $k \ge 5$.