On coloring graphs without induced forests

Abstract

The ℓ-Coloring problem is the problem to decide whether a graph can be colored with at most ℓ colors. Let Pk 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 P2 + P4. 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 = 2P3 is based on a novel structural result on the existence of small dominating sets in 2P3-free graphs that admit a k-coloring for some fixed k. © 2010 Springer-Verlag.