Fault-tolerant Pancyclicity Of Folded Hypercubes

The order of graph G is the number of vertices, denoted by v(G); and the girth of graph G is the length of the shortest cycle of graph G, denoted by g(G). A graph G is k-pancyclic if it contains cycles of every length from k to v(G) inclusive, and G is pancyclic if it is g(G)-pancyclic. G is edge-pancyclic if any fault-free edge lies on cycles of every length from g(G) to v(G) inclusive. A fault-free cycle is a cycle of no fault edge and no fault vertex. Let F be a faulty set of G, and fv, fe be the number of faulty vertices and edges of F, respectively. In this paper, we investigate the symmetry and fault-tolerant pancyclicity of folded hypercubes. We prove that folded hypercube FQn is symmetric, the even-dimension folded hypercubes contain fault free odd cycles of length l from n+1to2n-1, and that if there exist at most n-1fault elements in even-dimension folded hypercubes, then there exist fault-free odd cycles of length l from n+1to2n-2fv-1.