On graphs with no induced five-vertex path or paraglider

Article Type

Research Article

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Journal of Graph Theory


Given two graphs (Formula presented.) and (Formula presented.), a graph is (Formula presented.) -free if it contains no induced subgraph isomorphic to (Formula presented.) or (Formula presented.). For a positive integer (Formula presented.), (Formula presented.) is the chordless path on (Formula presented.) vertices. A paraglider is the graph that consists of a chorless cycle (Formula presented.) plus a vertex adjacent to three vertices of the (Formula presented.). In this paper, we study the structure of ((Formula presented.), paraglider)-free graphs, and show that every such graph (Formula presented.) satisfies (Formula presented.), where (Formula presented.) and (Formula presented.) are the chromatic number and clique number of (Formula presented.), respectively. Our bound is attained by the complement of the Clebsch graph on 16 vertices. More strongly, we completely characterize all the ((Formula presented.), paraglider)-free graphs (Formula presented.) that satisfies (Formula presented.). We also construct an infinite family of ((Formula presented.), paraglider)-free graphs such that every graph (Formula presented.) in the family has (Formula presented.). This shows that our upper bound is optimal up to an additive constant and that there is no (Formula presented.) -approximation algorithm for the chromatic number of ((Formula presented.), paraglider)-free graphs for any (Formula presented.).

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Open Access, Green

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