Anti-Ramsey Number of Friendship Graphs

An edge-colored graph is called $\mathit{\text{rainbow}}\mathit{\text{graph}}$ if all the colors on its edges are distinct. For a given positive integer $n$ and a family of graphs $\mathcal{G}$, the anti-Ramsey number $ar(n,\mathcal{G})$ is the smallest number of colors $r$ required to ensure that, no matter how the edges of the complete graph $K_n$ are colored using exactly $r$ colors, there will always be a rainbow copy of some graph $G$ from the family $\mathcal{G}$. A friendship graph $F_k$ is the graph obtained by combining $k$ triangles that share a common vertex. In this paper, we determine the anti-Ramsey number $ar(n,\{F_k\})$ for large values of $n$. Additionally, we also determine the $ar(n,\{K_{1,k},k{K}_2\}$, where $K_{1,k}$ is a star graph with $k+1$ vertices and $k{K}_2$ is a matching of size $k$.