Babu, J.Basavaraju, M.Sunil Chandran, L.S.Francis, M.C.2026-02-052019Discrete Applied Mathematics, 2019, 255, , pp. 109-1160166218Xhttps://doi.org/10.1016/j.dam.2018.08.004https://idr.nitk.ac.in/handle/123456789/24680Given 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 ?(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. © 2018 Elsevier B.V.Combinatorial mathematicsMathematical techniquesColourful pathGraph GInduced pathsTriangle-free graphsGraph theory