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

Given two graphs H1 and H2, a graph G is (H1,H2)-free if it contains no subgraph isomorphic to H1 or H2. Let Pt and Cs be the path on t vertices and the cycle on s vertices, respectively. In this paper we show that for any (P6, C4)-free graph G it holds that χ(G) ≤ 3 2 ω(G), where χ(G) and ω(G) are the chromatic number and clique number of G, respectively. Our bound is attained by several graphs, for instance, the five-cycle, the Petersen graph, the Petersen graph with an additional universal vertex, and all 4-critical (P6, C4)-free graphs other than K4 (see [18]). The new result unifies previously known results on the existence of linear χ-binding functions for several graph classes. Our proof is based on a novel structure theorem on (P6, C4)-free graphs that do not contain clique cutsets. Using this structure theorem we also design a polynomial time 3/2-approximation algorithm for coloring (P6, C4)-free graphs. Our algorithm computes a coloring with 3 2 ω(G) colors for any (P6, C4)-free graph G in O(nm) time.