Star Chromatic Index of Halin Graphs

TL;DR

This paper establishes a tight upper bound for the star chromatic index of Halin graphs, confirming a conjecture for cubic Halin graphs and advancing understanding of their edge coloring properties.

Contribution

It provides a tight upper bound for the star chromatic index of Halin graphs, confirming a conjecture for cubic cases.

Findings

Proves the bound $loor{rac{3 riangle}{2}}+2$ for all Halin graphs.

Confirms the conjecture for cubic Halin graphs.

Advances knowledge on star edge colorings of specific graph classes.

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.