2-edge-Hamilton-connected dragonfly network

The dragonfly networks are being used in the supercomputers of today. It is of interest to study the topological properties of dragonfly networks. Let $G=\left(V\right(G),E(G\left)\right)$ be a graph. Let X be a subset of $\{uv:u,v\in V(G)\text{and}u\ne v\}$ such that every component induced by X on $V\left(G\right)$ is a path. If, $\left|X\right|\le k$ and after adding all edges in X to G, the resulting graph contains a Hamiltonian cycle that includes all edges in X, then the graph G is called k-edge-Hamilton-connected. This property can be used to design and optimize routing and forwarding algorithms. By finding such Hamiltonian cycle containing specific edges in the network, it can be ensured that every node can act as an intermediate node to forward packets through a specific channel, thus enabling efficient data transmission and routing. For $k=2$, determining whether a graph is k-edge-Hamilton-connected is a challenging problem, as it is known to be NP-complete. 2-edge-Hamilton-connected is an extension of Hamilton-connected. In this paper, we prove that the relative arrangement dragonfly network, a type of dragonfly network constructed by the global connections based on relative arrangements, is 2-edge-Hamilton-connected, and this property shows that dragonfly networks have strong reliability. In addition, we determined that $D(n,h,g)$ is 1-Hamilton-connected and paired 2-disjoint path coverable with $n\ge 4$ and $h\ge 2$.