The number of 4-colorings of the Hamming cube

TL;DR

This paper proves that the number of proper 4-colorings of a d-dimensional hypercube asymptotically equals 6e times 2 to the power of 2^d, confirming a conjecture from 2012 using entropy and isoperimetric methods.

Contribution

It establishes the asymptotic count of 4-colorings of hypercubes, combining information theory and isoperimetric techniques in a novel way.

Findings

Number of 4-colorings asymptotically equals 6e * 2^{2^d}

Confirms a conjecture by Engbers and Galvin (2012)

Uses entropy and isoperimetric methods in proof

Abstract

Let $Q_d$ be the $d$-dimensional hypercube and $N=2^d$. We prove that the number of (proper) 4-colorings of $Q_d$ is asymptotically \[6e2^N,\] as was conjectured by Engbers and Galvin in 2012. The proof uses a combination of information theory (entropy) and isoperimetric ideas originating in work of Sapozhenko in the 1980's.