The Number of Colorings of the Middle Layers of the Hamming Cube

For an odd integer \(n = 2d-1\), let \({\mathcal {B}}_d\) be the subgraph of the hypercube \(Q_n\) induced by the two largest layers. In this paper, we describe the typical structure of proper q-colorings of \(V({\mathcal {B}}_d)\) and give asymptotics on the number of such colorings when q is an even number. The proofs use various tools including information theory (entropy), Sapozhenko’s graph container method and a recently developed method of Jenssen and Perkins that combines Sapozhenko’s graph container lemma with the cluster expansion for polymer models from statistical physics.