On k-detour subgraphs of hypercubes

A spanning subgraph G of a graph H is a k-detour subgraph of H if for each pair of vertices x, y ∈ V (H), the distance, distG(x, y), between x and y in G exceeds that in H by at most k. Such subgraphs sometimes also are called additive spanners. In this paper, we study k-detour subgraphs of the n-dimensional cube, Qn, with few edges or with moderate maximum degree. Let ∆(k, n) denote the minimum possible maximum degree of a k-detour subgraph of Qn. The main result is that for every k ≥ 2 and n ≥ 21, e−2k n lnn ≤ ∆(k, n) ≤ 20 n ln lnn lnn . On the other hand, for each fixed even k ≥ 4 and large n, there exists a k-detour subgraph of Qn with average degree at most 2 + 24−k/2 + o(1). ∗This material is based upon work supported by the National Science Foundation under Grant DMS0400498.