Star edge coloring of Cactus graphs

TL;DR

This paper proves a conjecture about the star chromatic index of Cactus graphs, a subclass of outerplanar graphs, showing it is bounded by a function of maximum degree.

Contribution

It confirms the conjecture for Cactus graphs, extending the understanding of star edge coloring in specific graph classes.

Findings

Star chromatic index of Cactus graphs is at most (3+1)

The conjecture holds for Cactus graphs, a subclass of outerplanar graphs

Provides bounds for star edge coloring in Cactus graphs

Abstract

A star edge coloring of a graph $G$ is a proper edge coloring of $G$ such that no path or cycle of length four is bi-colored. The star chromatic index of $G$, denoted by $\chi^{\prime}_{s}(G)$, is the minimum $k$ such that $G$ admits a star edge coloring with $k$ colors. Bezegov{\'a} et al. (Star edge coloring of some classes of graphs, J. Graph Theory, 81(1), pp.73-82. 2016) conjectured that the star chromatic index of outerplanar graphs with maximum degree $\Delta$, is at most $\left\lfloor\frac{3\Delta}{2}\right\rfloor+1$. In this paper, we prove this conjecture for a class of outerplanar graphs, namely Cactus graphs, wherein every edge belongs to at most one cycle.