Hypercube subgraphs with local detours

A minimal detour subgraph of the n-dimensional cube is a spanning subgraph G of Qn having the property that for vertices x, y of Qn, distances are related by dG(x; y) dQn(x; y) + 2. For a spanning subgraph G of Qn to be a local detour subgraph, we require only that the above inequality be satis ed whenever x and y are adjacent in Qn. Let f(n) (respectively, fl(n) ) denote the minimum number of edges in any minimal detour (respectively, local detour) subgraph of Qn (cf. Erd}os et al. [1]). In this paper we nd the asymptotics of fl(n) by showing that 3 2(1 O(n )) < fl(n) < 3 2(1 + o(1)): We also show that f(n) > 3:00001 2 (for n > n0); thus eventually fl(n) < f(n) answering a question of [1] in the negative. We nd the order of magnitude of Fl(n), the minimum possible maximum degree in a local detour subgraph of Qn : p 2n+ 0:25 0:5 Fl(n) 1:5 p 2n 1: This work was partially supported by a grant of DIMACS. DIMACS is a cooperative project of Rutgers University, Princeton University, AT&T Labs, Bell Labs and Bellcore. DIMACS is an NSF Science and Technology Center, founded under contract STC-91-19999; and also receives support from the New Jersey Commission on Science and Technology. This work was partially supported by a grant of DIMACS and by the grants 96-01-01614 and 97-01-01075 of the Russian Foundation for Fundamental Research.