Bipanconnectivity and Bipancyclicity in k-ary n-cubes

Abstract

In this paper we give precise solutions to problems posed by Wang, An, Pan, Wang and Qu and by Hsieh, Lin and Huang. In particular, we show that Qn k is bipanconnected and edge-bipancyclic, when k ges 3 and n ges 2, and we also show that when k is odd, Qn k is m-panconnected, for m=(n(k-1)+2k-6)/2, and (k-1)-pancyclic (these bounds are optimal). We introduce a path-shortening technique, called progressive shortening, and strengthen existing results, showing that when paths are formed using progressive shortening then these paths can be efficiently constructed and used to solve a problem relating to the distributed simulation of linear arrays and cycles in a parallel machine whose interconnection network is Qn k, even in the presence of a faulty processor.