阶图与边多面体的H-性质
On the H-property for step-graphons and edge polytopes
摘要：
图元$W$可以用作随机模型，对任意大的$n$的$n$节点上的图$G_n$进行采样。一个图子$W$被称为具有$H$ -性质，如果当$n$趋于无穷大时，$G_n$允许分解成概率为1的不相交圈。这样的分解称为哈密尔顿分解。本文给出了$H$ -性质成立的必要条件。证明建立在与$W$和$H$ -属性相关联的有限无向图的所谓边多面体之间的由此建立的连接上。建立在其属性，我们提供了一个纯粹的几何解决方案的随机图问题。更准确地说，我们分配两个自然对象的$W$，我们称之为浓度向量和骨架图，分别表示为$x^*$和$S$。然后我们建立了$H$ -性质成立的两个必要条件：（1）$S$的边多面体（记为$\mathcal{X}(S)$）是满秩的;（2）$x^* \in \mathcal{X}(S)$ .
英文摘要：
Graphons $W$ can be used as stochastic models to sample graphs $G_n$ on $n$ nodes for $n$ arbitrarily large. A graphon $W$ is said to have the $H$-property if $G_n$ admits a decomposition into disjoint cycles with probability one as $n$ goes to infinity. Such a decomposition is known as a Hamiltonian decomposition. In this paper, we provide necessary conditions for the $H$-property to hold. The proof builds upon a hereby established connection between the so-called edge polytope of a finite undirected graph associated with $W$ and the $H$-property. Building on its properties, we provide a purely geometric solution to a random graph problem. More precisely, we assign two natural objects to $W$, which we term concentration vector and skeleton graph, denoted by $x^*$ and $S$ respectively. We then establish two necessary conditions for the $H$-property to hold: (1) the edge-polytope of $S$, denoted by $\mathcal{X}(S)$, is of full rank, and (2) $x^* \in \mathcal{X}(S)$.
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