A New View of Hypercube Genus
Richard H. Hammack, Paul C. Kainen · American Mathematical Monthly · 2021
Beineke, Harary, and Ringel discovered a formula for the minimum genus of a torus in which the n-dimensional hypercube graph can be embedded. We give a new proof of the formula by building this surface as a union of certain faces in the hypercube’s 2-skeleton. For odd dimension n, the entire 2-skeleton decomposes into (n−1)/2 copies of the surface, and the intersection of any two copies is the hypercube graph.