Globally bi-3∗-connected graphs

AbstractA 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 1p-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 1p-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