Hamiltonian properties of some compound networks

Abstract 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 D V ( m , d , n ) be the compound networks of the disc-ring graph D ( m , d ) and the hypercube-like graphs H L n , and D H ( m , d , n ) be the compound networks of D ( m , d ) and H n which is the set of all ( n − 2 ) -fault Hamiltonian and ( n − 3 ) -fault Hamiltonian-connected graphs in H L n . We obtain that every graph in D V ( m , d , n ) is Hamiltonian which improves the known results that the D T c u b e , the D L c u b e and the D C c u b e are Hamiltonian obtained by Hung [Theoret. Comput. Sci. 498 (2013) 28-45]. Furthermore, we derive that D H ( m , d , n ) is ( n − 1 ) -edge-fault Hamiltonian. As corollaries, the ( n − 1 ) -edge-fault Hamiltonicity of the D R H L n including the D T ( m , d , n ) and the D C ( m , d , n ) is obtained. Moreover, the ( n − 1 ) -edge-fault Hamiltonicity of D H ( m , d , n ) is optimal.