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| Main Authors: | , , |
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| Format: | Preprint |
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2026
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| Online Access: | https://arxiv.org/abs/2605.25907 |
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| _version_ | 1866914599705509888 |
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| author | Ma, Menghan You, Lihua Zhang, Xiaoxue |
| author_facet | Ma, Menghan You, Lihua Zhang, Xiaoxue |
| contents | Let $\mathbf{G}=\{G_1,\dots,G_{n-1}\}$ be a collection of not necessarily distinct $n$-vertex graphs with the same vertex set $V$. A path $P$ with $V(P)\subseteq V$ and $|E(P)|\leq n-1$ is rainbow in $\mathbf{G}$, if there exists an injection $ϕ\colon E(P)\to [n-1]$ such that $e\in E(G_{ϕ(e)})$ for each $e\in E(P)$. The graph collection $\mathbf{G}$ is said to be \emph{rainbow panconnected} if for every pair of vertices $x,y\in V$, there exists a rainbow path of $k$ vertices joining $x$ and $y$ in $\mathbf{G}$ for every integer $k\in \left[d_{\mathbf{G}}(x,y)+1, n\right]$, where $d_{\mathbf{G}}(x,y)$ is the length of a shortest rainbow path between $x$ and $y$ in $\mathbf{G}$. In this paper, we study the rainbow panconnectivity of $\mathbf{G}$ under the minimum degree condition. Our result improves upon the corresponding results of [J. Graph Theory, \textbf{104}(2)(2023), 341--359] and [Electron. J. Combin., \textbf{32}(4)(2025), \#P4.17]. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_25907 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Rainbow panconnectivity in a graph collection Ma, Menghan You, Lihua Zhang, Xiaoxue Combinatorics 05C38 Let $\mathbf{G}=\{G_1,\dots,G_{n-1}\}$ be a collection of not necessarily distinct $n$-vertex graphs with the same vertex set $V$. A path $P$ with $V(P)\subseteq V$ and $|E(P)|\leq n-1$ is rainbow in $\mathbf{G}$, if there exists an injection $ϕ\colon E(P)\to [n-1]$ such that $e\in E(G_{ϕ(e)})$ for each $e\in E(P)$. The graph collection $\mathbf{G}$ is said to be \emph{rainbow panconnected} if for every pair of vertices $x,y\in V$, there exists a rainbow path of $k$ vertices joining $x$ and $y$ in $\mathbf{G}$ for every integer $k\in \left[d_{\mathbf{G}}(x,y)+1, n\right]$, where $d_{\mathbf{G}}(x,y)$ is the length of a shortest rainbow path between $x$ and $y$ in $\mathbf{G}$. In this paper, we study the rainbow panconnectivity of $\mathbf{G}$ under the minimum degree condition. Our result improves upon the corresponding results of [J. Graph Theory, \textbf{104}(2)(2023), 341--359] and [Electron. J. Combin., \textbf{32}(4)(2025), \#P4.17]. |
| title | Rainbow panconnectivity in a graph collection |
| topic | Combinatorics 05C38 |
| url | https://arxiv.org/abs/2605.25907 |