Panconnectivity, fault-tolerant hamiltonicity and hamiltonian-connectivity in alternating group graphs

Jwo et al. lNetworks 23 l1993r 315–326r introduced the alternating group graph as an interconnection network topology for computing systems. They showed that the proposed structure has many advantages over n-cubes and star graphs. For example, all alternating group graphs are hamiltonian-connected li.e., every pair of vertices in the graph are connected by a hamiltonian pathr and pancyclic li.e., the graph can embed cycles with arbitrary length with dilation 1r. In this article, we give a stronger result: all alternating group graphs are panconnected, that is, every two vertices x and y in the graph are connected by a path of length k for each k satisfying dlx, yr ≤ k ≤ vVv - 1, where dlx, yr denotes the distance between x and y, and vVv is the number of vertices in the graph. Moreover, we show that the r-dimensional alternating group graph AGr, r ≥ 4, is lr - 3r-vertex fault-tolerant Hamiltonian-connected and lr - 2r-vertex fault-tolerant hamiltonian. The latter result can be viewed as complementary to the recent work of Lo and Chen lIEEE Trans. Parallel and Distributed Systems 12 l2001r 209–222r, which studies the fault-tolerant hamiltonicity in faulty arrangement graphs. © 2004 Wiley Periodicals, Inc. NETWORKS, Vol. 44l4r, 302–310 2004