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

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, ${\chi }_{st}^{{}^{\mathrm{\prime }}}(G),$ 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 $\chi_{st}^{'} (G )\leq \lfloor\frac{3\Delta}{2}\rfloor+1$when$G$ is an outerplanar graph with maximum degree $\mathrm{\Delta }\ge 3.$ In this paper we obtained that ${\chi }_{st}^{{}^{\mathrm{\prime }}}(G)\le \mathrm{\Delta }+6$ 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 ${\chi }_{st}^{{}^{\mathrm{\prime }}}(G)\le 9.$