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    Globally bi-3βˆ—-connected graphs

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    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 kβˆ—-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=(BβˆͺW,E) is globally bi-3βˆ—-connected if there is a 3βˆ—-container C(x,y) between any pair of vertices {x,y} with x∈B and y∈W. Furthermore, G is hyper globally bi-3βˆ—-connected if it is globally bi-3βˆ—-connected and there exists a 3βˆ—-container C(x,y) in Gβˆ’{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 Gβˆ’e is Hamiltonian for any e∈E. A bipartite graph G=(BβˆͺW,E) is 1p-Hamiltonian if Gβˆ’{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-3βˆ—-connected graph is 1p-Hamiltonian and every globally bi-3βˆ—-connected graph is 1-edge Hamiltonian. We present some examples of hyper globally bi-3βˆ—-connected graphs, some globally bi-3βˆ—-connected graphs that are not hyper globally bi-3βˆ—-connected, and some examples of 1-edge Hamiltonian bipartite graphs that are not globally bi-3βˆ—-connected
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