A note on a conjecture of star chromatic index for 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, ${\chi }_{st}^{{}^{\prime }}(G),$ 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 ${\chi }_{st}^{{}^{\prime }}(G)\le \lfloor \frac{3\Delta }{2}\rfloor +1$ when $G$ is an outerplanar graph with maximum degree $\Delta \ge 3.$ In this paper we obtained that ${\chi }_{st}^{{}^{\prime }}(G)\le \Delta +6$ 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 ${\chi }_{st}^{{}^{\prime }}(G)\le 9.$