Star Chromatic Index of Halin Graphs

Abstract

A star edge coloring of a graph $G$ is a proper edge coloring of $G$ such that every path and cycle of length four in $G$ uses at least three different colors. The star chromatic index of $G$, is the smallest integer $k$ for which $G$ admits a star edge coloring with $k$ colors. In this paper, we obtain tight upper bound $\lfloor \frac{3\Delta }{2}\rfloor +2$ for the star chromatic index of every Halin graph, that proves the conjecture of Dvo{ř}{á}k et al. (J Graph Theory, 72 (2013), 313--326) for cubic Halin graphs.