Abstract: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 $\phi\colon E(P)\to [n-1]$ such that $e\in E(G_{\phi(e)})$ 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].
| Subjects: | Combinatorics (math.CO) |
| MSC classes: | 05C38 |
| Cite as: | arXiv:2605.25907 [math.CO] |
| (or arXiv:2605.25907v1 [math.CO] for this version) | |
| https://doi.org/10.48550/arXiv.2605.25907 arXiv-issued DOI via DataCite (pending registration) |
From: Xiaoxue Zhang [view email]
[v1]
Mon, 25 May 2026 14:42:49 UTC (13 KB)
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