Anti-Ramsey number of intersecting cliques

Abstract

An edge-colored graph is called a rainbow graph if all its edges have distinct colors. The anti-Ramsey number $ar(n,G)$, for a graph $G$ and a positive integer $n$, is defined as the minimum number of colors $r$ such that every exact $r$-edge-coloring of the complete graph $K_n$ contains at least one rainbow copy of $G$. A $(k,r)$-fan graph, denoted $F_{k,r}$, is a graph composed of $k$ cliques each of size $r$, all intersecting at exactly one common vertex. In this paper, we determine $ar(n,F_{k,r})$ for $n\ge 256r^{16}(k+1{)}^5$, $k\ge 1$, and $r\ge 2$.