Linearly χ-Bounding (P_6,C_4)-Free Graphs
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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 χ(G)<3/2ω(G), where χ(G) and ω(G) are the chromatic number and clique number of G, respectively. Petersen graph. 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 (P_6,C_4)-free graphs other than K_4 (see HH17). 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 (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 3/2ω(G) colors for any (P_6,C_4)-free graph G in O(n^2m) time.
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