  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 them. The proofs use various tools including
information theory (entropy), Sapozhenko's graph container method and a
recently developed method of M. Jenssen and W. Perkins that combines
Sapozhenko's graph container lemma with the cluster expansion for polymer
models from statistical physics.
