Vertex-deleted subgraphs and regular factors from regular graph

Let $k$, $m$ and $r$ be three integers such that $2\le k\le m\le r$. Let $G$ be a $2r$-regular, $2m$-edge-connected graph of odd order. We obtain some sufficient conditions, which guarantee $G-v$ contains a $k$-factor for all $v\in V(G)$.