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| Format: | Preprint |
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2026
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| Online Access: | https://arxiv.org/abs/2604.27535 |
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| _version_ | 1866910179802480640 |
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| author | Li, Luyi Wang, Yubo Yan, Guiying |
| author_facet | Li, Luyi Wang, Yubo Yan, Guiying |
| contents | Let $ n \in \mathbb{N} $ with $ n \geq 3 $, and let $\mathcal{G} = \{G_i:i\in [n]\} $ be a family of $ n $-vertex graphs on a common vertex set $V$, where the graphs in the family do not need to be distinct. A graph $H$ with vertex set $V$ is \emph{rainbow} in $\mathcal{G}$ if there exists an injection $ ϕ: E(H) \to [n] $ such that $e \in E(G_{ϕ(e)})$ for every edge $e \in E(H)$, where $|E(H)|\leq n$. In 2020, Joos and Kim proved that $\mathcal{G}$ contains a rainbow Hamiltonian cycle under the Dirac-type condition. Recently, Liu, Chen, and Ma generalized this result by replacing the Dirac-type condition with a more general Ore-type condition involving degree sums of non-adjacent vertices: If $σ(\mathcal{G}) \geq n$, then $\mathcal{G}$ contains a rainbow Hamiltonian cycle, where the Ore-type condition $σ(\mathcal{G})$ is defined as follows: $
σ(\mathcal{G}) = \min\{d_p(u) + d_q(v) \mid uv \notin E(G_i) \text{ for some } i \in [n] \text{ and for all } p, q \in [n]\}. $
In this paper, under the Ore-type condition, we show that either each vertex of $V$ is contained in a rainbow cycle of length $\ell$ for every $\ell\in[4,n]$, or $G_1=\cdots=G_n=K_{\frac{n}{2},\frac{n}{2}}$. As a corollary, we deduce the rainbow pancyclicity of $\mathcal{G}$, which
supports the famous meta-conjecture posed by Bondy. Furthermore, we prove rainbow vertex-pancyclicity of $\mathcal{G}$ under the Ore-type condition and provide an extremal graph family to show that the result is sharp. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2604_27535 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Pancyclicity in Graph Families with the Ore-Type Condition Li, Luyi Wang, Yubo Yan, Guiying Combinatorics 05C07, 05C38, 05C45 Let $ n \in \mathbb{N} $ with $ n \geq 3 $, and let $\mathcal{G} = \{G_i:i\in [n]\} $ be a family of $ n $-vertex graphs on a common vertex set $V$, where the graphs in the family do not need to be distinct. A graph $H$ with vertex set $V$ is \emph{rainbow} in $\mathcal{G}$ if there exists an injection $ ϕ: E(H) \to [n] $ such that $e \in E(G_{ϕ(e)})$ for every edge $e \in E(H)$, where $|E(H)|\leq n$. In 2020, Joos and Kim proved that $\mathcal{G}$ contains a rainbow Hamiltonian cycle under the Dirac-type condition. Recently, Liu, Chen, and Ma generalized this result by replacing the Dirac-type condition with a more general Ore-type condition involving degree sums of non-adjacent vertices: If $σ(\mathcal{G}) \geq n$, then $\mathcal{G}$ contains a rainbow Hamiltonian cycle, where the Ore-type condition $σ(\mathcal{G})$ is defined as follows: $ σ(\mathcal{G}) = \min\{d_p(u) + d_q(v) \mid uv \notin E(G_i) \text{ for some } i \in [n] \text{ and for all } p, q \in [n]\}. $ In this paper, under the Ore-type condition, we show that either each vertex of $V$ is contained in a rainbow cycle of length $\ell$ for every $\ell\in[4,n]$, or $G_1=\cdots=G_n=K_{\frac{n}{2},\frac{n}{2}}$. As a corollary, we deduce the rainbow pancyclicity of $\mathcal{G}$, which supports the famous meta-conjecture posed by Bondy. Furthermore, we prove rainbow vertex-pancyclicity of $\mathcal{G}$ under the Ore-type condition and provide an extremal graph family to show that the result is sharp. |
| title | Pancyclicity in Graph Families with the Ore-Type Condition |
| topic | Combinatorics 05C07, 05C38, 05C45 |
| url | https://arxiv.org/abs/2604.27535 |