Note on Whitney’s Theorem for $k$-connected

Abstract

In this paper we refine Whitney’s Theorem on $k$-connected graphs for $k\ge 3$. In particular we show the following: Let $G$ be a $k$-connected graph with $k\ge 3$. For any two distinct vertices $u$ and $v$ of $G$ there are $k$ internally vertex disjoint paths $P_1[u,v],P_2[u,v],\dots ,P_k[u,v]$ such that $G–V(P_i(u,v))$ is connected for $i=1,2,\dots ,k$, where $P_i(u,v)$ denotes the internal vertices of the path $P_i[u,v]$. Further one of the following properties holds:

1. $G–V(P_i[u,v])$ is connected for $i=1,2,3$.
2. $G–V(P_i[u,v])$ is connected for $i=1,2$ and $G–V(P_i[u,v])$ has exactly two connected components for $i=3,4,\dots ,k$.

In addition, some other properties will be proved.