Panconnectivity and edge-pancyclicity of multidimensional torus networks

In this paper, the (bi)panconnectivity and edge-(bi)pancyclicity of n-dimensional torus networks are investigated. An n-dimensional torus T=T(k1,k2,…,kn) with diameter m0=∑i=1n⌊ki/2⌋, where ki≥3 for i=1,2,…,n, is one of the most popular interconnection networks, the k-ary n-cube Qnk (=T(k,k,…,k)) is its special class. For any two vertices u and v in T, we determine the set ρ(u,v) of all lengths of (u,v)-paths in T by using path-shortening technique that can be used efficiently to construct the (u,v)-paths in the torus T. In particular, the following results are obtained: (1)  The torus T is bipanconnected and edge-bipancyclic; (2)  If some kj≥3 is odd and the other ki≥4 is even for every i≠j, then the torus T is m1-panconnected, where m1=(kj−1)/2+m0, and the bound m1 is optimal; (3) If both ki≥3 and kj≥3 are odd, then the torus T is m0-panconnected, and the bound m0 is optimal. (4)  If some kj is odd, let k be the minimum over all odd ki, then T is (k+1)-edge pancyclic and the bound k+1 is optimal if T≠Qnk, and T is h-edge-pancyclic and the bound h=max{k−1,3} is optimal otherwise. Our results strengthen and generalize existing results.