Rainbow panconnectivity in a graph collection

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 $|E(P)|\le n-1$ is called \emph{rainbow} in $\mathbf{G}$, if there exists an injection $\ddot{I}•:E(P)\to [n-1]$ such that $e\in E(G_{\ddot{I}•(e)})$ 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].