Decomposition of augmented cubes into regular connected pancyclic subgraphs

Abstract

In this paper, we consider the problem of decomposing the augmented cube AQn into two spanning, regular, connected and pancyclic subgraphs. We prove that for n ≥ 4 and 2n − 1 = n1 + n2 with n1, n2 ≥ 2, the augmented cube AQn can be decomposed into two spanning subgraphs H1 and H2 such that each Hi is ni-regular and ni-connected. Moreover, Hi is 4-pancyclic if ni ≥ 3.Interconnection networks play an important role in communication systems and parallel computing. Such a network is usually represented by a graph where vertices stand for its processors and edges for links between the processors. Network topology is a crucial factor for interconnection networks as it determines the performance of the networks. Many interconnection network topologies have been proposed in literature such as mesh, torus, hypercube and hypercube like structures. The n-dimensional hypercube Q n is a popular interconnection network topology. It is an n-regular, n-connected, vertex-transitive graph with 2 n vertices and has diameter n.In 2002, Chaudam and Sunitha [7] introduced a variant of the hypercube Q n called augmented cube AQ n . The graph AQ n is (2n − 1)-regular, (2n − 1)-connected, pancyclic and vertex-transitive on 2 n vertices. However, the diameter of AQ n is ⌈n/2⌉ which is almost half the diameter of Q n . Hence there is less delay in data transmission in the augmented cube network than the hypercube network. Many results have been obtained in the literature to prove that the augmented cube is a good candidate for computer network topology design; see [7,12,13,15,21].One of the central issue in evaluating a network is to study the embedding problem. It is said that a graph H can be embedded into a graph G if it is isomorphic to a subgraph of G and if so, while modeling a network with graph, we can apply existing algorithms for graph H to the graph G. Cycle networks are suitable for designing simple algorithms with low communication cost. Since some parallel applications, such as those in image