Edge-disjoint spanners in tori

A spanning subgraph S = (V , E ) of a connected graph G = (V , E) is an (x + c)-spanner if for any pair of vertices u and v, dS(u, v) ≤ dG(u, v) + c where dG and dS are the usual distance functions in G and S, respectively. The parameter c is called the delay of the spanner. We study edge-disjoint spanners in graphs in multi-dimensional tori. We show that each two-dimensional torus has a set of two edge-disjoint spanners of delay approximately the size of the smaller dimension. Moreover, we show that this delay is close to the best possible. In three-dimensional tori, we find a set of three edge-disjoint spanners with delay approximately the sum of the sizes of the two smaller dimensions when all dimensions are of even size. Surprisingly, we also find a set of two edge-disjoint spanners in three-dimensional tori of constant delay. In d-dimensional tori, we show that for any k ≤ d/5, there is a set of k edge-disjoint spanners with delay depending only on k and the size of the smaller k dimensions. Crown Copyright© 2008 Published by Elsevier B.V. All rights reserved.