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 FQn F Q n is a well-known variation of an n -dimensional hypercube Qn Q n which can be constructed from Qn Q n by adding an edge to every pair of vertices with complementary addresses. FQn F Q n for any odd n  is known to bipartite. In this paper, let FFv F F v and FFe F F e denote the sets of faulty vertices and faulty edges in FQn F Q n . Then, we consider the pancyclicity and bipancyclicity properties in FQn?FFv?FFe F Q n ? F F v ? F F e , as follows: For n?3 n ? 3 , FQn?FFv?FFe F Q n ? F F v ? F F e contains a fault-free cycle of every even length from 4 to 2n?2?|FFv| 2 n ? 2 ? | F F v | , where |FFv|+|FFe|?n?1 | F F v | + | F F e | ? n ? 1 ; For n?4 n ? 4 is even, FQn?FFv?FFe F Q n ? F F v ? F F e contains a fault-free cycle of every odd length from n+1 n + 1 to 2n?2?|FFv|?1 2 n ? 2 ? | F F v | ? 1 , where |FFv|+|FFe|?n?1 | F F v | + | F F e | ? n ? 1 .
pancyclicity bipancyclicity folded hypercubes fault-free interconnection networks fault-tolerant