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 Theorem that every properly coloured graph contains a colourful path on χ(G) vertices. We explore a conjecture that states that every properly coloured triangle-free graph G contains an induced colourful path on χ(G) vertices and prove its correctness when the girth of G is at least χ(G). Recent work on this conjecture by Gyárfás and Sárközy, and Scott and Seymour has shown the existence of a function f such that if χ(G)≥f(k), then an induced colourful path on k vertices is guaranteed to exist in any properly coloured triangle-free graph G.