Improved bounds for anti-Ramsey numbers of matchings in outerplanar graphs

Let $\mathcal{O}_n$ be the set of all maximal outerplanar graphs of order $n$. Let $ar(\mathcal{O}_n,F)$ denote the maximum positive integer $k$ such that $T\in \mathcal{O}_n$ has no rainbow subgraph $F$ under a $k$-edge-coloring of $T$. Denote by $M_k$ a matching of size $k$. In this paper, we prove that $ar(\mathcal{O}_n,M_k)\le n+4k-9$ for $n\ge3k-3$, which expressively improves the existing upper bound for $ar(\mathcal{O}_n,M_k)$. We also prove that $ar(\mathcal{O}_n,M_5)=n+4$ for all $n\ge 15$.