# On Induced Colourful Paths in Triangle-free Graphs

@article{Babu2017OnIC,
title={On Induced Colourful Paths in Triangle-free Graphs},
author={Jasine Babu and Manu Basavaraju and L. Sunil Chandran and Mathew C. Francis},
journal={Electron. Notes Discret. Math.},
year={2017},
volume={61},
pages={69-75}
}
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… Expand

#### Topics from this paper

On induced colourful paths in triangle-free graphs
• Computer Science, Mathematics
• Discret. Appl. Math.
• 2019
A conjecture is explored that states that every properly coloured triangle-free graph G contains an induced colourful path on χ ( G ) vertices and is proved to be correctness when the girth of G is at least φ ( G ), which is a corollary of the Gallai–Roy–Vitaver Theorem. Expand
Structure and colour in triangle-free graphs
• Computer Science, Mathematics
• Electron. J. Comb.
• 2021
It is proved that every properly coloured triangle-free graph of chromatic number $\chi$ contains a rainbow independent set of size $\lceil\frac12\chi\rceil$ which is sharp up to a factor $2$. Expand
Induced Subgraphs of Graphs with Large Chromatic Number IX: Rainbow Paths
• Mathematics, Computer Science
• Electron. J. Comb.
• 2017
It is proved that for all nonnegative integers k,s there exists c with the following property: for every vertex-colouring (not necessarily optimal) of G, some induced subgraph of G is an s-vertex path, and all its vertices have different colours. Expand
Variants of the Gy\arf\as-Sumner Conjecture: Oriented Trees and Rainbow Paths
• Mathematics, Computer Science
• 2021
Given a finite family F of graphs, we say that a graph G is “F-free” if G does not contain any graph in F as a subgraph. We abbreviate F-free to just “F -free” when F = {F}. A vertex-coloured graph HExpand