Vertex-fault-tolerant cycles embedding in 4-conditionally faulty folded hypercubes

Abstract

A network is said to be g-conditionally faulty if its every vertex has at least g fault-free neighbors, where g≥1. 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 be bipartite. In this paper, let FFv denote the set of faulty vertices in FQn, and let FFQn(e) denote the set of faulty vertices which are incident to the end-vertices of any fault-free edge e∈E(FQn). Then, under the 4-conditionally faulty and |FFQn(e)|≤n−3, we consider for the vertex-fault-tolerant cycles embedding properties in FQn−FFv, as follows: 1.For n≥4, FQn−FFv contains a fault-free cycle of every even length from 4 to 2n−2|FFv|, where |FFv|≤2n−7;2.For n≥4 being even, FQn−FFv contains a fault-free cycle of every odd length from n+1 to 2n−2|FFv|−1, where |FFv|≤2n−7.