Induced subgraphs of graphs with large chromatic number IX: Rainbow paths

Abstract

We prove that for all integers K, s >= 0 there exists c with the following property. Let G be a graph with clique number at most K and chromatic number more than c. Then for every vertex-colouring (not necessarily optimal) of G, some induced subgraph of G is an s-vertex path, and all its vertices have different colours. This extends a recent result of Gyarfas and Sarkozy (2016), who proved the same for graphs G with K = 2 and girth at least five.