Improved Complexity Results on k-Coloring P t -Free Graphs

A graph is H-free if it does not contain an induced subgraph isomorphic to H . We denote by Pk and Ck the path and the cycle on k vertices, respectively. In this paper, we prove that 4-COLORING is NP-complete for P7-free graphs, and that 5-COLORING is NP-complete for P6-free graphs. These two results improve all previous results on kcoloring Pt-free graphs, and almost complete the classification of complexity of k-COLORING Pt-free graphs for k ≥ 4 and t ≥ 1, leaving as the only missing case 4-COLORING P6-free graphs. We expect that 4-COLORING is polynomial time solvable for P6-free graphs; in support of this, we describe a polynomial time algorithm for 4-COLORING P6free graphs which are also P -free, where P is the graph obtained from C4 by adding a new vertex and making it adjacent to exactly one vertex on the C4.