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arXiv · 1604.06070

On Induced Colourful Paths in Triangle-free Graphs

Jasine Babu·Manu Basavaraju·L. Sunil Chandran·Mathew C. Francis
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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-Vitaver Theorem that every properly coloured graph contains a colourful path on $\chi(G)$ vertices. We explore a conjecture that states that every properly coloured triangle-free graph $G$ contains an induced colourful path on $\chi(G)$ vertices and prove its correctness when the girth of $G$ is at least $\chi(G)$. Recent work on this conjecture by Gy\'arf\'as and S\'ark\"ozy, and Scott and Seymour has shown the existence of a function $f$ such that if $\chi(G)\geq f(k)$, then an induced colourful path on $k$ vertices is guaranteed to exist in any properly coloured triangle-free graph $G$.

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Jasine Babu, Manu Basavaraju, L. Sunil Chandran, Mathew C. Francis. 2016-04-20. On Induced Colourful Paths in Triangle-free Graphs. https://arxiv.org/abs/1604.06070

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