The star edge coloring of cubic Halin graphs with star chromatic index $5$

TL;DR

This paper investigates the star chromatic index of cubic Halin graphs, establishing conditions under which this index equals 5, thus advancing understanding of edge coloring constraints in these graphs.

Contribution

It identifies specific classes of cubic Halin graphs with star chromatic index exactly 5, resolving part of an open problem in graph coloring.

Findings

Cubic Halin graphs with caterpillar or complete tree structures have star chromatic index 5.

Most cubic Halin graphs, except a special case, have star chromatic index at most 6.

The paper narrows down conditions for when the star chromatic index is exactly 5.

Abstract

The star chromatic index of a graph $G$, denoted by $\chi'_{st}(G) $, is the minimum number of colors needed to properly color the edges of $G$ such that no path or cycle of length four is bi-colored. Casselgren et al. and Hou et al. independently proved that the star chromatic index of a cubic Halin graph, except a special graph, is at most $6$. It remains an open problem to determine which of such graphs have star chromatic index $5$. In this paper, we show that if $G\ne N_{e_2}$ is a cubic Halin graph whose tree is a caterpillar or a complete tree, then $\chi'_{st}(G)=5$.