List star edge coloring of generalized Halin graphs

TL;DR

This paper investigates the list star edge coloring of generalized Halin graphs, establishing upper bounds on the list star chromatic index that depend on the graph's maximum degree and structure, with bounds proven to be sharp.

Contribution

It extends star edge coloring results to generalized Halin graphs, providing new upper bounds for their list star chromatic index based on structural parameters.

Findings

Upper bound for list star chromatic index when |C| ≠ 5

For Δ ≥ 13, the bound is at most ⌊3Δ/2⌋

The bounds are proven to be sharp

Abstract

A star $k$-edge coloring is a proper edge coloring such that there are no bichromatic paths or cycles of length four. The smallest integer $k$ such that $G$ admits a star $k$-edge coloring is the star chromatic index of $G$. Deng \etal \cite{MR2933839}, and Bezegov{\'a} \etal \cite{MR3431294} independently proved that the star chromatic index of a tree is at most $\lfloor \frac{3\Delta}{2} \rfloor$, and the bound is sharp. Han \etal \cite{MR3924408} strengthened the result to list version of star chromatic index, and proved that $\lfloor \frac{3\Delta}{2} \rfloor$ is also the sharp upper bound for the list star chromatic index of trees. A generalized Halin graph is a plane graph that consists of a plane embedding of a tree $T$ with $\Delta(T) \geq 3$, and a cycle $C$ connecting all the leaves of the tree such that $C$ is the boundary of the exterior face. In this paper, we prove that if $H := T \cup C$ is a generalized Halin graph with $|C| \neq 5$, then its list star chromatic index is at most \[ \max\left\{\left\lfloor\frac{\theta(T) + \Delta(T)}{2}\right\rfloor, 2 \left\lfloor\frac{\Delta(T)}{2}\right\rfloor + 7\right\}, \] where $\theta(T) = \max_{xy \in E(T)}\{d_{T}(x) + d_{T}(y)\}$. As a consequence, if $H$ is a (generalized) Halin graph with maximum degree $\Delta \geq 13$, then the list star chromatic index is at most $\lfloor \frac{3\Delta}{2} \rfloor$. Moreover, the upper bound for the list star chromatic index is sharp.