Improved complexity results on k-coloring Pt-free graphs

A graph is H-free if it does not contain an induced subgraph isomorphic to H. We denote by Pt and Ct the path and the cycle on t 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. The second result is the first NP-hardness result for k-COLORING P6-free graphs. These two results improve all previous NP-complete results on k-coloring 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 of 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 P6-free graphs which are also banner-free, where banner is the graph obtained from C4 by adding a new vertex and making it adjacent to exactly one vertex on the C4.