Rainbow path and minimum degree in properly edge colored graphs

Abstract

A rainbow path in a properly edge colored graph is a path in which all the edges are colored with distinct colors. Let G be a properly edge colored graph with minimum degree δ and let t be the maximum length of a rainbow path in G. In this paper, we show that $t \geqslant \left\lfloor {\frac{{3\delta }} {5}} \right\rfloor$. It is easy to see that there exist graphs for which t ≤ δ; with respect to some proper edge coloring. For example, δ-regular graphs of chromatic index δ are graphs for which t ≤ δ, with respect to their optimum proper edge coloring. We leave open the question of getting a lower bound as close to δ as possible.