Hypercube orientations with only two in-degrees

We consider the problem of orienting the edges of the n-dimensional hypercube so only two different in-degrees a and b occur. We show that this can be done, for two specified in-degrees, if and only if an obvious necessary condition holds. Namely, there exist non-negative integers s and t so that s + t = 2n and as + bt = n2n−1 . This is connected to a question arising from constructing a strategy for a “hat puzzle.” AMS 2010 subject classification: 05C20; 00A08; 94B05

• Type: Journal
• Year: 2011
• Where: JCT
• Authors: Joe Buhler, Steve Butler, Ronald L. Graham, Eric Tressler
• Keywords: JCT 2011, N-dimensional Hypercube, Non-negative Integers, Obvious Necessary Condition