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

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 FQnFQn is a well-known variation of an n  -dimensional hypercube QnQn which can be constructed from QnQn by adding an edge to every pair of vertices with complementary addresses. FQnFQn for any odd n   is known to bipartite. In this paper, let FFvFFv and FFeFFe denote the sets of faulty vertices and faulty edges in FQnFQn. Then, we consider the pancyclicity and bipancyclicity properties in FQn−FFv−FFeFQn−FFv−FFe, as follows:1.For n≥3n≥3, FQn−FFv−FFeFQn−FFv−FFe contains a fault-free cycle of every even length from 4 to 2n−2⋅|FFv|2n−2⋅|FFv|, where |FFv|+|FFe|≤n−1|FFv|+|FFe|≤n−1;2.For n≥4n≥4 is even, FQn−FFv−FFeFQn−FFv−FFe contains a fault-free cycle of every odd length from n+1n+1 to 2n−2⋅|FFv|−12n−2⋅|FFv|−1, where |FFv|+|FFe|≤n−1|FFv|+|FFe|≤n−1.