Polynomial Bounds for Chromatic Number. IV: A Near-polynomial Bound for Excluding the Five-vertex Path

Abstract

A graph G is H-free if it has no induced subgraph isomorphic to H. We prove that a P_5-free graph with clique number omega ge 3 has chromatic number at most omega ^{log _2(omega )}. The best previous result was an exponential upper bound (5/27)3^{omega }, due to Esperet, Lemoine, Maffray, and Morel. A polynomial bound would imply that the celebrated Erdős-Hajnal conjecture holds for P_5, which is the smallest open case. Thus, there is great interest in whether there is a polynomial bound for P_5-free graphs, and our result is an attempt to approach that.