Bulky subgraphs of the hypercube

Let Q(d) be the d-dimensional hypercube on 2(d) vertices, and let G be its induced subgraph. We say that G is simple-majority if \G\ > 2(d-1) and G is bulky if it is connected and meets every facet of Q(d). We show that every simple-majority G has a bulky subgraph. This was conjectured by Alon, Seymour and Thomas (A separator theorem for non-planar graphs, J. Am. Math. Soc. 3 (1990) 801-808). Further, we show that such a subgraph can be chosen on d + 1 vertices if d less than or equal to 5 and on fewer than 6 x 1.5(d-5) vertices if d greater than or equal to 6. (C) 2000 Academic Press.