2-Edge Hamiltonian connectedness: Characterization and results in data center networks

A graph G is 2-edge Hamiltonian connected if for any edge set $E\subseteq \{uv:u,v\in V(G),u\ne v\}$ with $\left|E\right|\le 2$, $G\cup E$ has a Hamiltonian cycle containing all edges of $E$, where $G\cup E$ 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.