Paths And Cycles In Cartesian Product Of Graphs

The Hypercube7,Cartesian product of paths and torus networks are three kinds of classical internet model.Path system and cycle system are one of the focus of the struc-ture and connectivity of internet.The Hypercube network is one of the most versatile and efficient networks for the architecture of parallel computes.It is well studied for the design of parallel algorithms and for simulations of other popular networks.The cycle networks are fundamental in parallel and distributed computing.They are suitable for local area networks and for designing the simple algorithms with low communication cost.Bipancyclicity of a network is an important factor in determining whether the network topology can simulate cycles of various lengths.The connectivity of a network is one of the important parameters to evaluate the reliability and fault tolerance of a network.In this paper,we firstly investigate the bipancyclicity and the existence of 3-regular subgraphs of Cartesian product of paths.And then we prove that an n-dimensional torus contains a spanning subgraphs which is k-regular,k-connected and bipancyclic.The traditional connectivity is often used as a measure of system reliability and fault tolerance.However,this measure has an obvious deficiency.Lin et.al introduced the structure connectivity and substructure connectivity,and we study further the structure connectivity of k-ary n-cube based on it.