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 $\left\lfloor\frac{3\Delta}{2}\right\rfloor+2$ for the star chromatic index of every Halin graph, that proves the conjecture of Dvo{\v{r}}{\'a}k et al. (J Graph Theory, 72 (2013), 313--326) for cubic Halin graphs.