Edge-fault-tolerant vertex-pancyclicity of augmented cubes

—The n-dimensional augmented cube, denoted as AQ n , a variation of the hypercube, possesses some properties superior to those of the hypercube. In this paper, we show that every vertex in AQ n lies on a fault-free cycle of every length from 4 to 2 n , even if there are up to 2n  3 link faults. We also show that this result is optimal. I. INTRODUCTION A graph G is a triple consisting of a vertex set V(G), an edge set E(G), and a relation that associates with each edge two vertices called its endpoints [26]. We usually use a graph to represent the topology of an interconnection network (network for short). The hypercube is one of the most versatile and efficient interconnection networks discovered to date for parallel computation. The hypercube is ideally suited to both special-purpose and general-purpose tasks, and can efficiently simulate many other same sized networks [18]. We usually use Q n to denote an n-dimensional hypercube. Many variants of the hypercube have been proposed. The augmented cube, recently proposed by Choudum and Sunitha [5], is one of such variations. An n-dimensional augmented cube AQ n can be formed as an extension of Q n by adding some links. For any positive integer n, AQ n is a vertex transitive, (2n  1)-regular, and (2n  1)-connected graph with 2 n vertices. AQ n retains all favorable properties of Q n since Q n  AQ n. Moreover, AQ n possesses some embedding properties that Q n does not. Previous works relating to the augmented hypercube Linear arrays and rings, two of the most fundamental networks for parallel and distributed computation, are suitable for developing simple algorithms with low communication costs. Many efficient algorithms designed based on linear arrays and rings for solving a variety of algebraic problems and graph problems can be found in [18]. The pancyclicity of a network represents its power of embedding rings of all possible lengths. A graph G is called m-pancyclic whenever G contains a cycle of each length l for m  l  |V(G)|. A 3-pancyclic graph is called pancyclic. The arrangement graph [7], the hypercomplete network [6], the WK-recursive network [10], the alternating group graph [17], and the hyper-de Bruijn networks [11] are all pancyclic. A graph G is m-vertex-pancyclic (respectively,