Pancyclicity and bipancyclicity of folded hypercubes with both vertex and edge faults

Abstract

A graph is said to be pancyclic if it contains cycles of every length from its girth to its order inclusive; and a bipartite graph is said to be bipancyclic if it contains cycles of every even length from its girth to its order. The pancyclicity or the bipancyclicity of a given network is an important factor in determining whether the network's topology can simulate cycles of various lengths. An n-dimensional folded hypercube FQn is a well-known variation of an n-dimensional hypercube Qn which can be constructed from Qn by adding an edge to every pair of vertices with complementary addresses. FQn for any odd n is known to bipartite. In this paper, let FFv and FFe denote the sets of faulty vertices and faulty edges in FQn. Then, we consider the pancyclicity and bipancyclicity properties in FQn−FFv−FFe, as follows:1.For n≥3, FQn−FFv−FFe contains a fault-free cycle of every even length from 4 to 2n−2⋅|FFv|, where |FFv|+|FFe|≤n−1;2.For n≥4 is even, FQn−FFv−FFe contains a fault-free cycle of every odd length from n+1 to 2n−2⋅|FFv|−1, where |FFv|+|FFe|≤n−1.