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. It is interesting to think of what analogous result one could obtain if one considers induced colourful paths instead of just colourful paths. We explore a conjecture that states that every properly coloured triangle-free graph G contains an induced colourful path on χ(G) vertices. As proving this conjecture in its fullest generality seems to be difficult, we study a special case of the conjecture. We show that the conjecture is true when the girth of G is equal to χ(G). Even this special case of the conjecture does not seem to have an easy proof: our method involves a detailed analysis of a special kind of greedy colouring algorithm. This result settles the conjecture for every properly coloured triangle-free graph G with girth at least χ(G).