Globally bi-3∗3∗-connected graphs

A kk-container  C(x,y)C(x,y) in a graph G=(V,E)G=(V,E) is a set of kk internally node-disjoint paths between vertices xx and yy. A k∗k∗-container  C(x,y)C(x,y) of GG is a kk-container such that every vertex of GG is incident with a certain path in C(x,y)C(x,y). A bipartite graph G=(B∪W,E)G=(B∪W,E) is globally bi  -3∗3∗-connected   if there is a 3∗3∗-container C(x,y)C(x,y) between any pair of vertices {x,y}{x,y} with x∈Bx∈B and y∈Wy∈W. Furthermore, GG is hyper globally bi  -3∗3∗-connected   if it is globally bi-3∗3∗-connected and there exists a 3∗3∗-container C(x,y)C(x,y) in G−{z}G−{z} for any three different vertices x,yx,y, and zz of the same partite set of GG. A graph G=(V,E)G=(V,E) is 1-edge Hamiltonian if G−eG−e is Hamiltonian for any e∈Ee∈E. A bipartite graph G=(B∪W,E)G=(B∪W,E) is 1p1p-Hamiltonian   if G−{x,y}G−{x,y} is Hamiltonian for any pair of vertices {x,y}{x,y} with x∈Bx∈B and y∈Wy∈W. In this paper, we prove that every hyper globally bi-3∗3∗-connected graph is 1p1p-Hamiltonian and every globally bi-3∗3∗-connected graph is 1-edge Hamiltonian. We present some examples of hyper globally bi-3∗3∗-connected graphs, some globally bi-3∗3∗-connected graphs that are not hyper globally bi-3∗3∗-connected, and some examples of 1-edge Hamiltonian bipartite graphs that are not globally bi-3∗3∗-connected.