A note on a conjecture of star chromatic index for outerplanar graphs

TL;DR

This paper investigates the star chromatic index of outerplanar graphs, providing new upper bounds for 2-connected graphs with small diameter or maximum degree, advancing understanding of coloring properties in such graphs.

Contribution

It establishes improved upper bounds for the star chromatic index of 2-connected outerplanar graphs with specific diameter and degree constraints.

Findings

For 2-connected outerplanar graphs with diameter 2 or 3, star chromatic index ≤ Δ+6.

For 2-connected outerplanar graphs with maximum degree 5, star chromatic index ≤ 9.

Supports conjecture relating star chromatic index to maximum degree in outerplanar graphs.

Abstract

A star edge coloring of a graph $G$ is a proper edge coloring of $G$ without bichromatic paths or cycles of length four. The it star chromatic index, $\chi_{st}^{'} (G ),$ of $G$ is the minimum number $k$ for which $G$ has a star edge coloring by $k$ colors. In \cite{LB}, L. Bezegov$\acute{a}$ et al. conjectured that $\chi_{st}^{'} (G )\leq \lfloor\frac{3\Delta}{2}\rfloor+1$ when $G$ is an outerplanar graph with maximum degree $\Delta \geq 3.$ In this paper we obtained that $\chi_{st}^{'}(G) \leq \Delta+6$ when $G$ is an 2-connected outerplanar graph with diameter 2 or 3. If $G$ is an 2-connected outerplanar graph with maximum degree 5, then $\chi_{st}^{'}(G) \leq 9.$