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arXiv · 2506.07115

New Bounds on the Anti-Ramsey Number of Independent Triangles

Hongliang Lu·Xinyue Luo·Xinxin Ma
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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 $n\geq 15k+57$.

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Hongliang Lu, Xinyue Luo, Xinxin Ma. 2025-06-08. New Bounds on the Anti-Ramsey Number of Independent Triangles. https://arxiv.org/abs/2506.07115

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