Anti-Ramsey number of intersecting cliques

TL;DR

This paper determines the anti-Ramsey number for intersecting clique structures called $(k, r)$-fan graphs in large complete graphs, providing exact values under specified conditions.

Contribution

It establishes the exact anti-Ramsey number for $(k, r)$-fan graphs in sufficiently large complete graphs, a new result in rainbow graph theory.

Findings

Exact anti-Ramsey number for $F_{k, r}$ in large $K_n$

Conditions on $n$, $k$, and $r$ for the result to hold

Advances understanding of rainbow structures in edge-colored graphs

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 $n \geq 256r^{16}(k+1)^5$, $k \geq 1$, and $r \geq 2$.