The optimal χ-bound for (P7,C4,C5)-free graphs

Abstract

In this paper, we give an optimal χ-binding function for the class of (P7,C4,C5)-free graphs. We show that every (P7,C4,C5)-free graph G has χ(G)≤⌈119ω(G)⌉. To prove the result, we use a decomposition theorem obtained in Cameron et al. (2020) [4] combined with careful inductive arguments and a nontrivial use of the König theorem for bipartite matching.