Globally bi--connected graphs

A k -container C ( x , y ) in a graph G = ( V , E ) is a set of k internally node-disjoint paths between vertices x and y . A k ∗ -container C ( x , y ) of G is a k -container such that every vertex of G is incident with a certain path in C ( x , y ) . A bipartite graph G = ( B ∪ W , E ) is globally bi- 3 ∗ -connected if there is a 3 ∗ -container C ( x , y ) between any pair of vertices { x , y } with x ∈ B and y ∈ W . Furthermore, G is hyper globally bi- 3 ∗ -connected if it is globally bi- 3 ∗ -connected and there exists a 3 ∗ -container C ( x , y ) in G − { z } for any three different vertices x , y , and z of the same partite set of G . A graph G = ( V , E ) is 1-edge Hamiltonian if G − e is Hamiltonian for any e ∈ E . A bipartite graph G = ( B ∪ W , E ) is 1 p -Hamiltonian if G − { x , y } is Hamiltonian for any pair of vertices { x , y } with x ∈ B and y ∈ W . In this paper, we prove that every hyper globally bi- 3 ∗ -connected graph is 1 p -Hamiltonian and every globally bi- 3 ∗ -connected graph is 1-edge Hamiltonian. We present some examples of hyper globally bi- 3 ∗ -connected graphs, some globally bi- 3 ∗ -connected graphs that are not hyper globally bi- 3 ∗ -connected, and some examples of 1-edge Hamiltonian bipartite graphs that are not globally bi- 3 ∗ -connected.