Babu, J.Basavaraju, M.Sunil Chandran, L.Francis, M.C.2026-02-052017Electronic Notes in Discrete Mathematics, 2017, 61, , pp. 69-7515710653https://doi.org/10.1016/j.endm.2017.06.022https://idr.nitk.ac.in/handle/123456789/25553Given 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 Gyárfás and Sárközy, 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.Colourful PathInduced PathTriangle-free Graph