On Coloring Graphs without Induced Forests

The l-COLORING problem is the problem to decide whether a graph can be colored with at most E colors. Let P-k denote the path on k vertices and G H and 2H the disjoint union of two graphs G and H or two copies of H, respectively. We solve a known open problem by showing that 3-COLORING is polynomial-time solvable for the class of graphs with no induced 2P(3). This implies that the complexity of 3-COLORING for graphs with no induced graph H is now classified for any fixed graph H on at most 6 vertices. The VERTEX COLORING problem is the problem to determine the chromatic number of a graph. We show that VERTEX COLORING is polynomial-time solvable for the class of triangle-free graphs with no induced 2P3 and for the class of triangle-free graphs with no induced P-2 + P-4. This solves two open problems of Dabrowski, Lozin, Raman and Ries and implies that the complexity of VERTEX COLORING for triangle-free graphs with no induced graph H is now classified for any fixed graph H on at most 6 vertices. Our proof technique for the case H = 2P(3) is based on a novel structural result on the existence of small dominating sets in 2P(3)-free graphs that admit a k-coloring for some fixed k.