Stability for the Anti-Ramsey Number of Matchings

Let $n,r,s$ be three positive integers such that $n\ge 2s+5$. Let $K_r$ denote the complete graph of order $r$. Given a graph $F$, the anti-Ramsey number $ar(n,F)$ is defined as the minimum number $C$ such that any edge-coloring of $K_n$ with exactly $C$ colors contains a rainbow copy of $F$. Let $H$ be an edge-colored graph on $K_n$ with at least $g(n,s)$ colors, where $$g(n,s)=\max \{\left(\genfrac{}{}{0px}{}{n}{2}\right)-\left(\genfrac{}{}{0px}{}{n-s+1}{2}\right)+5,\left(\genfrac{}{}{0px}{}{2s-1}{2}\right)+n+1\}\mathrm{.}$$ In this paper, we establish a stability type result for the anti-Ramsey number of matchings. Specifically, if $H$ does not have a rainbow matching of size $s+2$, then $H$ contains either a monochromatic complete graph $K_{n-s}$ or a monochromatic $K_{n-2s-1}\vee \overline{K_{2s+1}}$.