Anti-Ramsey Number of Friendship Graphs

TL;DR

This paper determines the minimum number of colors needed in edge-colorings of complete graphs to guarantee rainbow friendship graphs and other specific subgraphs, advancing understanding of anti-Ramsey numbers for these structures.

Contribution

It explicitly calculates the anti-Ramsey numbers for large n for friendship graphs and certain other graph families, filling gaps in combinatorial graph theory.

Findings

Calculated $ar(n, \{F_k\})$ for large n.

Determined $ar(n, \{K_{1,k}, kK_2\})$ for large n.

Extended anti-Ramsey theory to new classes of graphs.

Abstract

An edge-colored graph is called \textit{rainbow 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 $ar(n, \{K_{1,k}, kK_2\}$, where $K_{1,k}$ is a star graph with $ k+1$ vertices and $kK_2$ is a matching of size $k$.