Connectivity preserving spanning $(u,v)$-paths in $k$-connected graphs

Abstract

Hasunuma [Graphs Combin. 41:10 (2025)] proved that for $k\ge 2$, there exists a function $f(k)=O(k)$ such that every $k$-connected graph $G$ of order $n\ge f(k)$ with $\hat{I}´(G)\ge \frac{n}{2}$ contains a Hamiltonian cycle $H$ such that $G-E(H)$ is $k$-connected. In this paper, we show that for $k\ge 2$, if $G$ is a $k$-connected graph of order $n\ge 6k+6$ with minimum degree at least $\frac{n+1}{2}$, then for any two distinct vertices $u,v\in V(G)$, there exists a Hamiltonian $(u,v)$-path $P$ such that $G-E(P)$ is $k$-connected. Moreover, we further extend this result to $s$ internally disjoint spanning $(u,v)$-paths.