Abstract
AbstractGiven two graphs and , a graph is ‐free if it contains no induced subgraph isomorphic to or . Let and be the path on vertices and the cycle on vertices, respectively. In this paper we show that for any ‐free graph it holds that , where and are the chromatic number and clique number of , respectively. Our bound is attained by several graphs, for instance, the 5‐cycle, the Petersen graph, the Petersen graph with an additional universal vertex, and all ‐critical ‐free graphs other than (see Hell and Huang [Discrete Appl. Math. 216 (2017), pp. 211–232]). The new result unifies previously known results on the existence of linear ‐binding functions for several graph classes. Our proof is based on a novel structure theorem on ‐free graphs that do not contain clique cutsets. Using this structure theorem we also design a polynomial time ‐approximation algorithm for coloring ‐free graphs. Our algorithm computes a coloring with colors for any ‐free graph in time.
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