On induced colourful paths in triangle-free graphs

Abstract 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 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 Gyarfas and Sarkozy, 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 .