Three Complexity Results on Coloring $P_k$-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 $P_6$-ree 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