The anti-Ramsey number of $M_5$ in outerplanar graphs

Let $\mathcal{O}_n$ denote the family of all maximal outerplanar graphs on $n$ vertices. Given an outerplanar graph $H$, let $ar_{\mathcal{_O}}(n,H)$ denote the maximum number of colors $k$ such that a $k$-edge coloring of a maximal outerplanar graph $T\in \mathcal{O}_n$ that contains no rainbow copy of $H$. Let $M_k$ denote the matching of size $k$. In this paper, we prove that $ar_{\mathcal{_O}}(n,M_5)=n+4$ for $n\ge 12$, which improves one of the results of Jin and Ye (2018).