Panconnectivity and Pancyclicity of Hypercube-Like Interconnection Networks with Faulty Elements

In this paper, we deal with the graph G0⊕G1 obtained from merging two graphs G0 and G1 with n vertices each by n pairwise nonadjacent edges joining vertices in G0 and vertices in G1. The main problems studied are how fault-panconnectivity and fault-pancyclicity of G0 and G1 are translated into fault-panconnectivity and fault-pancyclicity of G0⊕G1, respectively. Many interconnection networks such as hypercubelike interconnection networks can be represented in the form of G0 ⊕G1 connecting two lower dimensional networks G0 and G1. Applying our results to a class of hypercube-like interconnection networks called restricted HL-graphs, we show that in a restricted HL-graph G of degree m(≥ 3), each pair of vertices are joined by a path in G\F of every length from 2m − 3 to |V (G\F )| − 1 for any set F of faulty elements (vertices and/or edges) with |F | ≤ m− 3, and there exists a cycle of every length from 4 to |V (G\F )| for any fault set F with |F | ≤ m− 2.