Hypercube subgraphs with local detours

A minimal detour subgraph of the n-dimensional cube is a spanning subgraph G of Q(n) having the property that, for vertices x, y of Q(n), distances are related by d(G)(x, y) less than or equal to d(Qn)(x, y)+2. For a spanning subgraph G of Q(n) to be a local detour subgraph, we require only that the above inequality be satisfied whenever x and y are adjacent in Q(n). Let f(n) (respectively, f(l)(n)) denote the minimum number of edges in any minimal detour (respectively, local detour) subgraph of Q(n) (cf. Erdos et al. [1]). In this article, we find the asymptotics of fi(n) by showing that 3.2(n)(1 - O(n(-1/2))) < fi (n) < 3.2(n)(1 + o(1)). We also show that f(n) > 3.00001.2(n) Vbr n > no), thus eventually fi(n) < f(n), answering a question of II] in the negative. We find the order of magnitude of FI(n), the minimum possible maximum degree in a local detour subgraph of Q(n) : root 2n+0.25- 0.5 less than or equal to Fl (n) less than or equal to 1.5 root 2n - 1. (C) 1999 John Wiley & Sons, Inc.