Title: Levelable graphs Show Chinese title

Title: 可分级图

Abstract: We study a family of positive weighted well-covered graphs, which we call levelable graphs, that are related to a construction of level artinian rings in commutative algebra. A graph $G$ is levelable if there exists a weight function with positive integer values on the vertices of $G$ such that $G$ is well-covered with respect to this weight function. That is, the sum of the weights in any maximal independent set of vertices of $G$ is the same. We describe some of the basic properties of levelable graphs and classify the levelable graphs for some families of graphs, e.g., trees, cubic circulants, Cameron--Walker graphs. We also explain the connection between levelable graphs and a class of level artinian rings. Applying a result of Brown and Nowakowski about weighted well-covered graphs, we show that for most graphs, their edge ideals are not Cohen--Macaulay. Show Chinese abstract

Abstract: 我们研究了一类正权值的良复盖图，我们称这类图为可分级图，它们与交换代数中Artin环的分级构造有关。 一个图 $G$ 是可分级的，当且仅当在 $G$ 的顶点上存在一个取正整数值的权函数，使得 $G$ 关于此权函数是良复盖的。 也就是说，在 $G$ 的任意极大独立顶点集中，权重之和是相同的。 我们描述了可分级图的一些基本性质，并对某些图族（例如树、三次循环图、Cameron–Walker 图）进行了可分级图的分类。 我们还解释了可分级图与一类分级Artin环之间的联系。 利用Brown和Nowakowski关于加权良复盖图的结果，我们证明对于大多数图，它们的边理想不是Cohen–Macaulay的。