Generalized shuffle-exchange digraphs: Hamiltonian properties

The k-ary n-dimensional shuffle-exchange directed network S(k, n) consists of k(n) nodes such that each node is represented as an k-ary n-tuple vector, a(1)a(2) ... a(n), where a(i) is in [0,k - 1]. Node a(1)a(2) ... a(n) is adjacent to node a(2)a(3) ... a(n)a(1) (one shuffle link) and k - 1 other nodes a(1)a(2) ... a(n-1)b, where b is an element of [0,k - 1] and b not equal a(n) (k - 1 exchange links). S(k, n) have been widely used as topologies for VLSI networks, parallel architectures, and communication systems. However, since S(k, n) does not exist for the number of nodes between k(n) and k(n+1), Liu and Hsu have recently proposed a class of digraphs GS(k, N), which generalized S(k, n) to any number N of nodes. They also showed that GS(li, N) retains all the nice properties of S(li, n). In this paper, we survey these combinatorial properties and study Hamiltonian properties of GS(lc, N). In particular, we have successfully obtained Hamiltonian circuit for GS(k, k(k + 1)) for any k > 2.