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arXiv · 2304.03203

The number of colorings of the middle layers of the Hamming cube

Lina Li·Gweneth McKinley·Jinyoung Park
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Abstract

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.

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Lina Li, Gweneth McKinley, Jinyoung Park. 2023-05-25. The number of colorings of the middle layers of the Hamming cube. https://arxiv.org/abs/2304.03203

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