Three complexity results on coloring -free graphs

We prove three complexity results on vertex coloring problems restricted to P k -free graphs, i.e., graphs that do not contain a path on k vertices as an induced subgraph. First of all, we show that the pre-coloring extension version of 5-coloring remains NP-complete when restricted to P 6 -free graphs. Recent results of Hoàng et al. imply that this problem is polynomially solvable on P 5 -free graphs. Secondly, we show that the pre-coloring extension version of 3-coloring is polynomially solvable for P 6 -free graphs. This implies a simpler algorithm for checking the 3-colorability of P 6 -free graphs than the algorithm given by Randerath and Schiermeyer. Finally, we prove that 6-coloring is NP-complete for P 7 -free graphs. This problem was known to be polynomially solvable for P 5 -free graphs and NP-complete for P 8 -free graphs, so there remains one open case.