Hamiltonian properties of some compound networks

Given two regular graphs G and H, the compound graph of G and H is constructed by replacing each vertex of G by a copy of H and replacing each link of G by a link which connects corresponding two copies of H. Let DV(m,d,n) be the compound networks of the disc-ring graph D(m,d) and the hypercube-like graphs HLn, and DH(m,d,n) be the compound networks of D(m,d) and Hn which is the set of all (n−2)-fault Hamiltonian and (n−3)-fault Hamiltonian-connected graphs in HLn. We obtain that every graph in DV(m,d,n) is Hamiltonian which improves the known results that the DTcube, the DLcube and the DCcube are Hamiltonian obtained by Hung [Theoret. Comput. Sci. 498 (2013) 28-45]. Furthermore, we derive that DH(m,d,n) is (n−1)-edge-fault Hamiltonian. As corollaries, the (n−1)-edge-fault Hamiltonicity of the DRHLn including the DT(m,d,n) and the DC(m,d,n) is obtained. Moreover, the (n−1)-edge-fault Hamiltonicity of DH(m,d,n) is optimal.