The optimal chromatic bound for even-hole-free graphs without induced seven-vertex paths
Abstract
The class of even-hole-free graphs has been extensively studied on its own and on its relation to perfect graphs. In this paper, we study the -boundedness of even-hole-free graphs which itself is an important topic in graph theory. In particular, we prove that every even-hole-free graph G without induced 7-vertex paths satisfies (G) 54ω(G), where (G) and ω(G) denote the chromatic number and clique number of G, respectively. This bound is optimal. Our result strictly extends the result of Karthick and Maffary KM19 on even-hole-free graphs without induced 6-vertex paths, and implies that even-hole-free graphs without induced 7-vertex paths satisfy Reed's Conjecture. Our proof relies on a heavy structural analysis on a maximal substructure called a nice blowup of a five-cycle and can be viewed for graphs in which all holes are of length five (graphs with all holes having the same length gain increasing interest in recent years COOK202496). Our result gives a partial answer to a conjecture of Wang and Wu WW25 on graphs in which all holes are of length 5. One of the key technical ingredients is a technical lemma proved via clique cutset argument combined with the idea of Infinite Descent Method (often used in number theory).