On induced colourful paths in triangle-free graphs

Authors

Jasine Babu, Indian Institute of Technology Palakkad
Manu Basavaraju, National Institute of Technology Karnataka
L. Sunil Chandran, Indian Institute of Science
Mathew C. Francis, Indian Statistical Institute Chennai

Article Type

Research Article

Publication Title

Discrete Applied 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–Vitaver 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

109

Last Page

116

DOI

10.1016/j.dam.2018.08.004

Publication Date

2-28-2019

Comments

Open Access, Green

Recommended Citation

Babu, Jasine; Basavaraju, Manu; Chandran, L. Sunil; and Francis, Mathew C., "On induced colourful paths in triangle-free graphs" (2019). Journal Articles. 945.
https://digitalcommons.isical.ac.in/journal-articles/945