Embedding the n-cube in lower dimensions (Q581416)

The sphericity of a graph G, sph(G), is the smallest integer m such that G is embeddable into the Euclidean space \(E^ m\) in a way that adjacent vertices have distance at most 1 while any two non-adjacent vertices are at distance more than 1. It is proved that for the n-dimensional cube \(Q_ n\) it holds \[ (1+o(1))n/\log_ 2n<sph(Q_ n)<(2\quad \log_ 23+o(1))n/\log_ 2n. \]