Hamiltonicity of the WK-recursive network with and without faulty nodes

Recently, the WK-recursive network has received much attention due to its many favorable properties such as a high degree of scalability. By K(d,t), we denote the WK-recursive network of level t, each of whose basic modules is a d-node complete graph, where d>1 and t≥1. In this paper, we first show that K(d,t) is Hamiltonian-connected, where d≥4. A network is Hamiltonian-connected if it contains a Hamiltonian path between every two distinct nodes. In other words, a Hamiltonian-connected network can embed the longest linear array between any two distinct nodes with dilation, congestion, load, and expansion all equal to one. Then, we construct fault-free Hamiltonian cycles in K(d,t) with at most d-3 faulty nodes, where d≥4. Since the connectivity of K(d,t) is d-1, the result is optimal.