Tight minimum colored degree condition for rainbow connectivity
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866915019577360384 |
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| author | Czygrinow, Andrzej Yuan, Xiaofan |
| author_facet | Czygrinow, Andrzej Yuan, Xiaofan |
| contents | Let $G = (V, E)$ be a graph on $n$ vertices, and let $c: E \to P$, where $P$ is a set of colors. Let $δ^c(G) = \min_{v \in V} \{ d^{c}(v) \}$ where $d^c(v)$ is the number of colors on edges incident to a vertex $v$ of $G$. In 2011, Fujita and Magnant showed that if $G$ is a graph on $n$ vertices that satisfies $δ^c(G)\geq n/2$, then for every two vertices $u, v$ there is a properly-colored $u,v$-path in $G$. In this paper, we show that the same bound for $δ^c(G)$ implies that any two vertices are connected by a rainbow path. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_09095 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Tight minimum colored degree condition for rainbow connectivity Czygrinow, Andrzej Yuan, Xiaofan Combinatorics 05C35, 05C15 Let $G = (V, E)$ be a graph on $n$ vertices, and let $c: E \to P$, where $P$ is a set of colors. Let $δ^c(G) = \min_{v \in V} \{ d^{c}(v) \}$ where $d^c(v)$ is the number of colors on edges incident to a vertex $v$ of $G$. In 2011, Fujita and Magnant showed that if $G$ is a graph on $n$ vertices that satisfies $δ^c(G)\geq n/2$, then for every two vertices $u, v$ there is a properly-colored $u,v$-path in $G$. In this paper, we show that the same bound for $δ^c(G)$ implies that any two vertices are connected by a rainbow path. |
| title | Tight minimum colored degree condition for rainbow connectivity |
| topic | Combinatorics 05C35, 05C15 |
| url | https://arxiv.org/abs/2411.09095 |