Anti-Ramsey Number of Intersecting Odd Cycles

Abstract

For a graph $H$, the anti-Ramsey number $\operatorname{ar}(n,H)$ is the maximum number of colors in an edge-coloring of $K_n$ containing no rainbow copy of $H$, where a copy is rainbow if its edges have pairwise distinct colors. Let $s,t$ be nonnegative integers with $s+t\ge2$, and let $H_{s,t}$ be a graph consisting of $s$ triangles and $t$ odd cycles of fixed lengths at least $5$, all sharing exactly one common vertex and otherwise pairwise vertex-disjoint. Liu et al. (2024) determined $\operatorname{ar}(n,H_{s,0})$ for $s\ge3$ and $n\ge50s^2$. In this paper, we determine the exact value of $\operatorname{ar}(n,H_{s,t})$ for every fixed $H_{s,t}$ with $t\ge1$ and all sufficiently large $n$.