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arXiv · 2509.25949

Anti-Ramsey Numbers for Spanning Linear Forests of 3-Vertex Paths and Matchings

Ali Ghalavand·Xueliang Li
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Abstract

A subgraph in an edge-colored graph is called rainbow if all its edges have distinct colors. For a graph $G$ and an integer $n$, the anti-Ramsey number $AR(n,G)$ is the maximum number of colors in an edge-coloring of $K_n$ that contains no rainbow copy of $G$. We study $AR(n, kP_3 \cup tP_2)$, where $kP_3 \cup tP_2$ is the linear forest of $k$ disjoint paths on three vertices and a matching of size $t$. Recently, Jie and Jin [Discrete Appl. Math. 386 (2026) 30-57] determined this number for $k\geq 2$, $t\geq\frac{k^2-3k+4}{2}$ and $n=2t+3k$. Here we solve the spanning case $n=3k+2t$ for all $k\ge1$, $t\ge2$ with no extra restrictions.

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Ali Ghalavand, Xueliang Li. 2025-09-30. Anti-Ramsey Numbers for Spanning Linear Forests of 3-Vertex Paths and Matchings. https://arxiv.org/abs/2509.25949

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