On the construction of all shortest vertex-disjoint paths in Cayley graphs of abelian groups

Abstract

Cayley graphs provide a group-theoretic model for designing and analyzing symmetric interconnection networks. In this paper, we give a sufficient condition for the existence of m vertex-disjoint shortest paths from one source vertex to other m (not necessarily distinct) destination vertices in a Cayley graph of an abelian group, where m≤n and n is the cardinality of a (group) generator of the abelian group. In addition, when the condition holds, the m vertex-disjoint shortest paths can be constructed in O(mn) time, which is optimal in the worst case when O(n) ≤ the order of diameter. By applying our results, we can easily obtain the necessary and sufficient conditions, which can be verified in nearly optimal time, for the existence of all shortest vertex-disjoint paths in generalized hypercubes and odd tori. In the situation that all of the source vertex and destination vertices are mutually distinct, brute-force computations show that the probability of the existence of the m vertex-disjoint shortest paths in an r-dimensional generalized hypercube with r coordinates each of order k is greater than 94%, 70%, 96%, 99%, and 99% for (k,r,m)=(2,7,7),(3,4,8),(4,3,6),(5,3,6), and (6,3,6), respectively.