On induced colourful paths in triangle-free graphs

Given a graph G = (V, E) whose vertices have been properly coloured, we say that a path in G is colourful if no two vertices in the path have the same colour. It is a corollary of the Gallai-Roy-Vitaver Theorem that every properly coloured graph contains a colourful path on chi(G) vertices. We explore a conjecture that states that every properly coloured triangle free graph G contains an induced colourful path on chi(G) vertices and prove its correctness when the girth of G is at least chi(G). Recent work on this conjecture by Gyarfas and Sarkozy, and Scott and Seymour has shown the existence of a function f such that if chi(G) >= f (k), then an induced colourful path on k vertices is guaranteed to exist in any properly coloured triangle-free graph G. (C) 2018 Elsevier B.V. All rights reserved.