Show that if any $k+1$ vertices of $k-$connected graph with at least 3 vertices span at least one-edge, then the graph is hamiltonian.
I know that a graph $G$ is said to be $k-$connected if there does not exist a set of $k-1$ vertices whose removal disconnects the graph. Also, a graph is hamiltonian if it contains a hamiltonian cycle which is a cycle that visits every vertex exactly once.
This problem has been getting to me for the past two days and I would really appreciate some assistance in solving this. It is coming from the book Modern Graph Theory by Bella Ballobas.
Many thanks in advance for your time and suggestions.