Stability for the Anti-Ramsey Number of Matchings
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866911588869472256 |
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| author | Zhang, Xuechun Lu, Hongliang |
| author_facet | Zhang, Xuechun Lu, Hongliang |
| contents | Let $n, r, s$ be three positive integers such that $n\geq 2s+5$. Let $K_r$ denote the complete graph of order $r$. Given a graph $F$, the anti-Ramsey number $ar(n,F)$ is defined as the minimum number $C$ such that any edge-coloring of $K_n$ with exactly $C$ colors contains a rainbow copy of $F$. Let $H$ be an edge-colored graph on $K_n$ with at least $g(n,s)$ colors, where \[
g(n,s)=\max\left\{ \binom{n}{2} - \binom{n - s + 1}{2} + 5, \binom{2s - 1}{2} + n + 1 \right\}. \] In this paper, we establish a stability type result for the anti-Ramsey number of matchings. Specifically, if $H$ does not have a rainbow matching of size $s+2$, then $H$ contains either a monochromatic complete graph $K_{n-s}$ or a monochromatic $K_{n - 2s - 1} \vee \overline{K_{2s + 1}}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_11505 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Stability for the Anti-Ramsey Number of Matchings Zhang, Xuechun Lu, Hongliang Combinatorics Let $n, r, s$ be three positive integers such that $n\geq 2s+5$. Let $K_r$ denote the complete graph of order $r$. Given a graph $F$, the anti-Ramsey number $ar(n,F)$ is defined as the minimum number $C$ such that any edge-coloring of $K_n$ with exactly $C$ colors contains a rainbow copy of $F$. Let $H$ be an edge-colored graph on $K_n$ with at least $g(n,s)$ colors, where \[ g(n,s)=\max\left\{ \binom{n}{2} - \binom{n - s + 1}{2} + 5, \binom{2s - 1}{2} + n + 1 \right\}. \] In this paper, we establish a stability type result for the anti-Ramsey number of matchings. Specifically, if $H$ does not have a rainbow matching of size $s+2$, then $H$ contains either a monochromatic complete graph $K_{n-s}$ or a monochromatic $K_{n - 2s - 1} \vee \overline{K_{2s + 1}}$. |
| title | Stability for the Anti-Ramsey Number of Matchings |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2604.11505 |