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

TL;DR

This paper establishes improved upper bounds for the anti-Ramsey numbers of matchings in outerplanar graphs, providing exact values for certain cases and advancing understanding of rainbow subgraph avoidance.

Contribution

It introduces tighter bounds for anti-Ramsey numbers of matchings in outerplanar graphs, including an exact formula for the case when the matching size is five.

Findings

Proved that $ar(\

ext{O}_n, M_k) \

ext{le } n+4k-9$ for $n \\ge 3k-3$; an improved upper bound.

Abstract

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 $n\ge 15$.