线图子的H-性质
The H-property of Line Graphons
摘要：
我们探讨在本文中的充分条件的$H$ -属性举行，特别是所谓的线graphons的重点。图子是从单位平方$[0,1]^2$到闭区间$[0,1]$的对称可测函数。图子可以用来对随机图进行采样，如果从图子采样的$n$节点上的图允许由不相交的圈覆盖的节点--这样的覆盖被称为Hamilton分解--几乎肯定是$n \to \infty$，则图子被称为具有$H$ -属性。阶梯图子是指在定义域中的矩形上分段为常数的图子。对于一个阶梯形，我们分配两个对象：它的浓度向量，编码矩形的面积，和它的阶梯形图，描述它们的支持。这两个对象在我们早期的工作[3]中被用来建立阶跃图介子具有$H$ -性质的必要条件。在本文中，我们证明了这些条件对于线图子类本质上也是充分的，即，其骨架图是在结束节点处具有自环的线图的阶图子。我们还研究既不满足必要条件也不满足充分条件的边界情况。
英文摘要：
We explore in this paper sufficient conditions for the $H$-property to hold, with a particular focus on the so-called line graphons. A graphon is a symmetric, measurable function from the unit square $[0,1]^2$ to the closed interval $[0,1]$. Graphons can be used to sample random graphs, and a graphon is said to have the $H$-property if graphs on $n$ nodes sampled from it admit a node-cover by disjoint cycles -- such a cover is called a Hamiltonian decomposition -- almost surely as $n \to \infty$. A step-graphon is a graphon which is piecewise constant over rectangles in the domain. To a step-graphon, we assign two objects: its concentration vector, encoding the areas of the rectangles, and its skeleton-graph, describing their supports. These two objects were used in our earlier work [3] to establish necessary conditions for a step-graphon to have the $H$-property. In this paper, we prove that these conditions are essentially also sufficient for the class of line-graphons, i.e., the step-graphons whose skeleton graphs are line graphs with a self-loop at an ending node. We also investigate borderline cases where neither the necessary nor the sufficient conditions are met.
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