Linearly $\chi$-Bounding $(P_6,C_4)$-Free Graphs

Abstract

Given two graphs $H_1$ and $H_2$, a graph G is $(H_1,H_2)$-free if it contains no subgraph isomorphic to $H_1$ or $H_2$. Let $P_t$ and $C_s$ be the path on t vertices and the cycle on s vertices, respectively. In this paper we show that for any $(P_6,C_4)$-free graph G it holds that $\chi (G)\le \frac{3}{2}\omega (G)$, where $\chi (G)$ and $\omega (G)$ are the chromatic number and clique number of G, respectively. Our bound is attained by $C_5$ and the Petersen graph. The new result unifies previously known results on the existence of linear $\chi$-binding functions for several graph classes. Our proof is based on a novel structure theorem on $(P_6,C_4)$-free graphs that do not contain clique cutsets. Using this structure theorem we also design a polynomial time 3/2-approximation algorithm for coloring $(P_6,C_4)$-free graphs. Our algorithm computes a coloring with $\frac{3}{2}\omega (G)$ colors for any $(P_6,C_4)$-free graph G in $O(n^2m)$ time.