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

IF 1 2区数学 Q1 MATHEMATICS

Combinatorica Pub Date : 2025-01-02 DOI:10.1007/s00493-024-00128-w

Lina Li, Gweneth McKinley, Jinyoung Park

引用次数: 0

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 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.

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汉明立方体中间层的着色数

对于一个奇整数\(n = 2d-1\)，设\({\mathcal {B}}_d\)为由两个最大层引起的超立方体\(Q_n\)的子图。本文描述了\(V({\mathcal {B}}_d)\)的真q染色的典型结构，并给出了当q为偶数时真q染色个数的渐近性。证明使用了各种工具，包括信息论（熵），Sapozhenko的图容器方法以及Jenssen和Perkins最近开发的一种方法，该方法将Sapozhenko的图容器引理与统计物理中聚合物模型的簇展开相结合。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Combinatorica 数学-数学

CiteScore

1.90

自引率

0.00%

发文量

审稿时长

>12 weeks

期刊介绍： COMBINATORICA publishes research papers in English in a variety of areas of combinatorics and the theory of computing, with particular emphasis on general techniques and unifying principles. Typical but not exclusive topics covered by COMBINATORICA are - Combinatorial structures (graphs, hypergraphs, matroids, designs, permutation groups). - Combinatorial optimization. - Combinatorial aspects of geometry and number theory. - Algorithms in combinatorics and related fields. - Computational complexity theory. - Randomization and explicit construction in combinatorics and algorithms.