Ore's Theorem for rainbow Hamiltonian-connected graphs
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866912761844334592 |
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| author | Li, Yupei Luo, Ruth |
| author_facet | Li, Yupei Luo, Ruth |
| contents | Let $G = (G_1, G_2, \ldots, G_m)$ be a collection of $m$ graphs on a common vertex set $V$. For a graph $H$ with vertices in $V$, we say that $G$ contains a rainbow $H$ if there is an injection $c: E(H) \to [m]$ such that for every edge $e \in E(H)$, we have $e \in E(G_{c(e)})$.
In this paper, we show that if $G = (G_1, \ldots, G_n)$ is a collection of graphs on $n$ vertices such that for every $i \in [n]$, $d_{G_i}(u) + d_{G_i}(v) \geq n$ whenever $uv \notin E(G_i)$, then either $G$ contains rainbow Hamiltonian paths between every pair of vertices, or $G$ contains a rainbow Hamiltonian cycle. Moreover, we prove a stronger version in which we may also embed prescribed rainbow linear forests into the Hamiltonian paths. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_12143 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Ore's Theorem for rainbow Hamiltonian-connected graphs Li, Yupei Luo, Ruth Combinatorics Let $G = (G_1, G_2, \ldots, G_m)$ be a collection of $m$ graphs on a common vertex set $V$. For a graph $H$ with vertices in $V$, we say that $G$ contains a rainbow $H$ if there is an injection $c: E(H) \to [m]$ such that for every edge $e \in E(H)$, we have $e \in E(G_{c(e)})$. In this paper, we show that if $G = (G_1, \ldots, G_n)$ is a collection of graphs on $n$ vertices such that for every $i \in [n]$, $d_{G_i}(u) + d_{G_i}(v) \geq n$ whenever $uv \notin E(G_i)$, then either $G$ contains rainbow Hamiltonian paths between every pair of vertices, or $G$ contains a rainbow Hamiltonian cycle. Moreover, we prove a stronger version in which we may also embed prescribed rainbow linear forests into the Hamiltonian paths. |
| title | Ore's Theorem for rainbow Hamiltonian-connected graphs |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2512.12143 |