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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 Theorem that every properly coloured graph contains a colourful path on $\chi(G)$ 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 $\chi(G)$ 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 $\chi(G)$. 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 $\chi(G)$.