A graph G is 2-edge Hamiltonian connected if for any edge set E⊆{uv:u,v∈V(G),u≠v} with |E|≤2, G∪E has a Hamiltonian cycle containing all edges of E, where G∪E is the graph obtained from G by including all edges of E. The problem of determining whether a graph is 2-edge Hamiltonian connected is challenging, as it is known to be NP-complete. This property is stronger than Hamiltonian connectedness, which indicates the existence of a Hamiltonian path between every pair of vertices in a graph. This paper first provides a characterization and a sufficiency for 2-edge Hamiltonian connectedness. Through this, we shed light on the fact that many well-known networks are 2-edge Hamiltonian connected, including BCube data center networks and some variations of hypercubes, and so on. Additionally, we demonstrate that DCell data center networks and Cartesian product graphs containing almost all generalized hypercubes are 2-edge Hamiltonian connected.

中文翻译：

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2 边缘哈密顿连通性：数据中心网络的表征和结果


如果对于任何边将 E⊆{uv：u，v∈V（G），u≠v} 设置为 |E|≤2，G∪E 有一个包含 E 的所有边的哈密顿循环，其中 G∪E 是从 G 包含所有边得到的图。确定图形是否为 2 边哈密顿量连接的问题具有挑战性，因为众所周知它是 NP 完备的。此属性比哈密顿连通性更强，哈密顿连通性表示图形中每对顶点之间存在哈密顿路径。本文首先提供了 2 边哈密顿连通性的特征和充分性。通过这一点，我们揭示了许多众所周知的网络都是 2 边哈密顿量连接的事实，包括 BCube 数据中心网络和超立方体的一些变体等。此外，我们还证明了 DCell 数据中心网络和包含几乎所有广义超立方体的笛卡尔乘积图是 2 边哈密顿连接。