Panconnectivity and pancyclicity of hypercube-like interconnection networks with faulty elements

Abstract

In this paper, we deal with the graph G 0 ⊕ G 1 obtained from merging two graphs G 0 and G 1 with n vertices each by n pairwise nonadjacent edges joining vertices in G 0 and vertices in G 1 . The main problems studied are how fault-panconnectivity and fault-pancyclicity of G 0 and G 1 are translated into fault-panconnectivity and fault-pancyclicity of G 0 ⊕ G 1 , respectively. Many interconnection networks such as hypercube-like interconnection networks can be represented in the form of G 0 ⊕ G 1 connecting two lower dimensional networks G 0 and G 1 . 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 2 m − 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 .