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 ln n (k, n) 20 n ln ln n ln n . 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. 1 

• Type: Journal
• Year: 2008
• Where: JGT
• Authors: Nana Arizumi, Peter Hamburger, Alexandr V. Kostochka
• Keywords: JGT 2008, K-detour Subgraph, Maximum Degree, Moderate Maximum Degree