Bulky Subgraphs of the Hypercube

Let Qdbe the d -dimensional hypercube on 2dvertices, and let G be its induced subgraph. We say that G is simple-majority if |G | > 2d − 1and G is bulky if it is connected and meets every facet of Qd. 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 ifd ≤ 5 and on fewer than 6 × 1.5d − 5vertices if d ≥ 6.