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arXiv 1101.0840math.CO
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H-染色环面

H-coloring tori

John Engbers, David Galvin

备注：
29 pages, some corrections and minor revisions from earlier version, this version to appear in Journal of Combinatorial Theory Series B
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摘要：

对于图$G$和$H$，$G$的$H$ -染色是从$G$的顶点到$H$的顶点的保持邻接的函数。$H$ -着色编码图论概念，如独立集和适当的着色，并且是统计物理学中硬约束模型研究的自然环境。本文研究了偶离散环面${\mathbb Z}^d_m$的$H$ -染色集，即顶点集${0, ..., m-1}^d$（$m$偶）上的图，如果两个相邻的字符串在一个坐标上相差1（mod $m$），而在其他坐标上都一致.这是一个二分图，具有二分类${\mathcal E}$和${\mathcal O}$。在$m=2$的情况下，偶数离散环面是离散超立方体或汉明立方体$Q_d$，${0,1}^d$上通常的最近邻图。对任何$H$和固定的$m$，我们得到了${\mathbb Z}^d_m$的$H$ -着色空间的结构特征.我们表明，它可以被划分成一个特殊的子集可以忽略不计的大小（$d$增长）和集合的子集索引的某些对$(A,B) \in V(H)^2$，每个$H$着色的子集索引的$(A,B)$有所有，但一个消失的比例的顶点从${\mathcal E}$映射到顶点从$A$，和所有，但一个消失的比例的顶点从${\mathcal O}$映射到顶点从$B$。这意味着对于$m$固定和$d$增长的${\mathbb Z}^d_m$的均匀选择的$H$着色的长程相关现象。我们的证明是通过分析一致选择的$H$ -染色的熵来进行的，并且推广了Kahn关于$m=2$和$H$是双无限路的特殊情形的方法，我们的结果推广到了$H$ -染色的自然加权模型.

英文摘要：

For graphs $G$ and $H$, an $H$-coloring of $G$ is a function from the vertices of $G$ to the vertices of $H$ that preserves adjacency. $H$-colorings encode graph theory notions such as independent sets and proper colorings, and are a natural setting for the study of hard-constraint models in statistical physics. We study the set of $H$-colorings of the even discrete torus ${\mathbb Z}^d_m$, the graph on vertex set ${0, ..., m-1}^d$ ($m$ even) with two strings adjacent if they differ by 1 (mod $m$) on one coordinate and agree on all others. This is a bipartite graph, with bipartition classes ${\mathcal E}$ and ${\mathcal O}$. In the case $m=2$ the even discrete torus is the discrete hypercube or Hamming cube $Q_d$, the usual nearest neighbor graph on ${0,1}^d$. We obtain, for any $H$ and fixed $m$, a structural characterization of the space of $H$-colorings of ${\mathbb Z}^d_m$. We show that it may be partitioned into an exceptional subset of negligible size (as $d$ grows) and a collection of subsets indexed by certain pairs $(A,B) \in V(H)^2$, with each $H$-coloring in the subset indexed by $(A,B)$ having all but a vanishing proportion of vertices from ${\mathcal E}$ mapped to vertices from $A$, and all but a vanishing proportion of vertices from ${\mathcal O}$ mapped to vertices from $B$. This implies a long-range correlation phenomenon for uniformly chosen $H$-colorings of ${\mathbb Z}^d_m$ with $m$ fixed and $d$ growing. Our proof proceeds through an analysis of the entropy of a uniformly chosen $H$-coloring, and extends an approach of Kahn, who had considered the special case of $m=2$ and $H$ a doubly infinite path. All our results generalize to a natural weighted model of $H$-colorings.

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