Cycle Embedding in Enhanced Hypercubes with Faulty Vertices

Abstract

The enhanced hypercube is a well-known variant of the hypercube and can be constructed from a hypercube by adding an edge to every pair of vertices with complementary addresses. Let Fv denote the set of faulty vertices in an n-dimensional enhanced hypercube Qn,k(1≤k≤n−1). In this paper, we conclude that if n≥2, then every fault-free edge of Qn,k−Fv lies on a fault-free cycle of every even length from 4 to 2n−2|Fv|, and if n(≥2) and k have the different parity, then every fault-free edge of Qn,k−Fv lies on a fault-free cycle of every possible odd length from n−k+4 to 2n−2|Fv|−1, where |Fv|≤n−2.