Improved Complexity Results on $k$-Coloring $P_t$-Free Graphs

TL;DR

This paper advances the understanding of the computational complexity of k-coloring P_t-free graphs by establishing NP-completeness results for specific cases and proposing a polynomial-time algorithm for a particular subclass.

Contribution

It proves NP-completeness of 4-COLORING for P_7-free graphs and 5-COLORING for P_6-free graphs, nearly completing the classification for k-coloring P_t-free graphs.

Findings

NP-completeness of 4-COLORING for P_7-free graphs

NP-completeness of 5-COLORING for P_6-free graphs

Polynomial-time algorithm for 4-COLORING P_6-free graphs that are also P-free

Abstract

A graph is $H$-free if it does not contain an induced subgraph isomorphic to $H$. We denote by $P_k$ and $C_k$ the path and the cycle on $k$ vertices, respectively. In this paper, we prove that 4-COLORING is NP-complete for $P_7$-free graphs, and that 5-COLORING is NP-complete for $P_6$-free graphs. These two results improve all previous results on $k$-coloring $P_t$-free graphs, and almost complete the classification of complexity of $k$-COLORING $P_t$-free graphs for $k\ge 4$ and $t\ge 1$, leaving as the only missing case 4-COLORING $P_6$-free graphs. We expect that 4-COLORING is polynomial time solvable for $P_6$-free graphs; in support of this, we describe a polynomial time algorithm for 4-COLORING $P_6$-free graphs which are also $P$-free, where $P$ is the graph obtained from $C_4$ by adding a new vertex and making it adjacent to exactly one vertex on the $C_4$.