Embedding Every Cycle in Conditionally Faulty Folded Hypercubes

Abstract

A network is said to be conditionally faulty if its every vertex is incident to 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. In this paper, we define that a network is said to be g-conditionally faulty if its every vertex is incident to at least g fault-free neighbors, and let FFv (respectively, FFe) denote the set of faulty vertices (respectively, faulty edges) in FQn. Then, we consider for the cycles embedding properties in FQn−FFv−FFe with 4-conditionally faulty, 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 |FFn|+|FFe| ≤ 2n−5; (2) For even n≥4, FQn−FFv−FFe contains a fault-free cycle of every odd length from n+1 to 2n−2|FFv|−1, where |FFv|+|FFe| ≤ 2n−5.