Hypercube orientations with only two in-degrees

TL;DR

This paper characterizes when it is possible to orient the edges of an n-dimensional hypercube so that only two in-degrees occur, linking the problem to a combinatorial condition and a hat puzzle strategy.

Contribution

The paper provides a complete characterization of hypercube orientations with only two in-degrees, establishing necessary and sufficient conditions based on combinatorial parameters.

Findings

Orientation exists if and only if the necessary condition involving s and t holds.

The problem is connected to a strategy for a hat puzzle.

Provides a combinatorial framework for hypercube edge orientations.

Abstract

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 $s+t=2^n$ and $as+bt=n2^{n-1}$. This is connected to a question arising from constructing a strategy for a "hat puzzle."