An Improved Degree Condition for Connectivity-Preserving Spanning $(u,v)$-Paths

Teng and Tian proved the following result. Let $k\ge 2$ and $t\ge 3$, and let $G$ be a $k$-connected graph of order $n$. If $n\ge 6k+1$ and $\delta (G)\ge \lceil (n+6)\mathrm{/}2\rceil$ when $t=3$, while $n\ge 6k+7t-17$ and $\delta (G)\ge \lceil (n+t+2)\mathrm{/}2\rceil$ when $t\ge 4$, then, for any two distinct vertices $u,v$ and every integer $s$ with $1\le s\le t$, there exist $s$ internally vertex-disjoint $(u,v)$-paths $P_1,\dots ,P_s$ whose union spans $G$ and such that $G-E(P_1\cup \cdots \cup P_s)$ is $k$-connected. They asked whether the minimum-degree condition could be lowered to $\delta (G)\ge \lceil (n+t)\mathrm{/}2\rceil$ for every $t\ge 3$. We answer this question affirmatively and further reduce the required order to $n\ge \max \{6k+9-3t,2k+t+3\}$.