Babu, J.Basavaraju, M.Sunil, Chandran, L.Francis, M.C.2020-03-312020-03-312017Electronic Notes in Discrete Mathematics, 2017, Vol.61, , pp.69-75https://idr.nitk.ac.in/handle/123456789/12361Given 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 ?(G) vertices. We explore a conjecture that states that every properly coloured triangle-free graph G contains an induced colourful path on ?(G) vertices and prove its correctness when the girth of G is at least ?(G). Recent work on this conjecture by Gy rf s and S rk zy, and Scott and Seymour has shown the existence of a function f such that if ?(G)?f(k), then an induced colourful path on k vertices is guaranteed to exist in any properly coloured triangle-free graph G. 2017 Elsevier B.V.Article