New Bounds on the Anti-Ramsey Number of Independent Triangles

TL;DR

This paper improves the lower bound on the number of vertices needed to determine the anti-Ramsey number for multiple disjoint triangles in complete graphs, extending previous results.

Contribution

It extends the known bounds for the anti-Ramsey number of vertex-disjoint triangles by increasing the minimum order of the complete graph.

Findings

Improved the lower bound on n to 15k+57 for the anti-Ramsey number of k disjoint triangles.

Extended previous results by Wu et al. on anti-Ramsey numbers.

Provided tighter bounds for rainbow triangle configurations in complete graphs.

Abstract

An edge-colored graph is called \textit{rainbow graph} if all the colors on its edges are distinct. Given a positive integer $n$ and a graph $G$, the \textit{anti-Ramsey number} $ar(n,G)$ is defined to be the minimum number of colors $r$ such that there exists a rainbow copy of $G$ in any exactly $r$-edge-coloring of $K_n$. Wu et al. (Anti-Ramsey numbers for vertex-disjoint triangles, \emph{Discrete. Math.}, \textbf{346} (2022), 113123) determined the anti-Ramsey number $ar(n, kK_3)$ for $n\geq 2k^2-k+2 $. In this paper, we extend this result by improving the lower bound on $n$ to $n\geq 15k+57$.