Sparse Hypercube 3-spanners

A t-spanner of a graph G=(V,E), is a sub-graph SG=(V,E′), such that E′⊆E and for every edge {u,v}∈E, there is a path from u to v in SG of length at most t. A minimum-edge t-spanner of a graph G, SG′, is the t-spanner of G with the fewest edges. For general graphs and for t=2, the problem of determining for a given integer s, whether |E(SG′)|⩽s is NP-Complete (Peleg and Schaffer, J. Graph Theory 13(1) (1989) 99–116). Peleg and Ullman (SIAM J. Comput. 18(4) (1989) 740–747), give a method for constructing a 3-spanner of the n-vertex Hypercube with fewer than 7n edges. In this paper we give an improved construction giving a 3-spanner of the n-vertex Hypercube with fewer than 4n edges and we present a lower bound of 3n/2−o(1) on the size of the optimal Hypercube 3-spanner.