New Bounds on the Anti-Ramsey Number of Independent Triangles

An edge-colored graph is called $\mathit{\text{rainbow}}\mathit{\text{graph}}$ if all the colors on its edges are distinct. Given a positive integer $n$ and a graph $G$, the $\mathit{\text{anti-Ramsey}}\mathit{\text{number}}$ $ar(n,G)$ is defined to be the minimum number of colors $r$ such that there exists a rainbow copy of $G$ in any exactly $r$-edge-coloring of $K_n$. Wu et al. (Anti-Ramsey numbers for vertex-disjoint triangles, \emph{Discrete. Math.}, $\mathbf{\text{346}}$ (2022), 113123) determined the anti-Ramsey number $ar(n,k{K}_3)$ for $n\ge 2k^2-k+2$. In this paper, we extend this result by improving the lower bound on $n$ to $n\ge 15k+57$.