On Induced Colourful Paths in Triangle-free Graphs

Article Type

Research Article

Publication Title

Electronic Notes in Discrete Mathematics

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 χ(G) vertices. We explore a conjecture that states that every properly coloured triangle-free graph G contains an induced colourful path on χ(G) vertices and prove its correctness when the girth of G is at least χ(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 χ(G)≥f(k), then an induced colourful path on k vertices is guaranteed to exist in any properly coloured triangle-free graph G.

First Page

69

Last Page

75

DOI

10.1016/j.endm.2017.06.022

Publication Date

8-1-2017

Comments

Open Access, Green

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