Contractible Edges of $k$-Connected Graphs for $k=4,5$

Abstract

Dean $([3])$ shows that if $G$ be a $k$-connected graph such that any fragment whose neighborhood contains an edge has cardinality exceeding $\frac{k}{2}$, then the subgraph $H=(V(G),E_k(G))$ formed by $V(G)$ and the $k$-contractible edges of $G$ is $2$-connected. In this paper, we show that for $k=4$, Dean’s result holds when reduced $\frac{k}{2}$ to $\frac{k}{4}$. But for $k\ge 5$, we give a counterexample to show that it is false and give a lower bound of the number of $k$-contractible edges for $k=5$.