论文标题
小图的Cohen-Macaulay二项式边缘理想
Cohen-Macaulay binomial edge ideals of small graphs
论文作者
论文摘要
cohen-Macaulay二项式边缘理想的特征的组合特性长期以来难以捉摸。最近的猜想将二项式边缘的Cohen-MacAulays与其基础图$ G $的特殊断开顶点的特殊连接集，称为\ textit {cut sets}。更确切地说，猜想指出，$ j_g $是Cohen-Macaulay，并且仅当$ j_g $未被混合并且收集$ g $的$ g $的集合是一个可访问的集合系统。在本文中，我们在理论上证明了所有图表的猜想，这些图形最高为$ 12 $顶点，并开发了一种算法，该算法允许计算所有图表的猜想，该猜想的所有图形最多可达$ 15 $的顶点，所有带有质量的块带有块最多为11美元的质量的质量。这显着扩大了以前的计算结果。
A combinatorial property that characterizes Cohen-Macaulay binomial edge ideals has long been elusive. A recent conjecture ties the Cohen-Macaulayness of a binomial edge ideal $J_G$ to special disconnecting sets of vertices of its underlying graph $G$, called \textit{cut sets}. More precisely, the conjecture states that $J_G$ is Cohen-Macaulay if and only if $J_G$ is unmixed and the collection of the cut sets of $G$ is an accessible set system. In this paper we prove the conjecture theoretically for all graphs with up to $12$ vertices and develop an algorithm that allows to computationally check the conjecture for all graphs with up to $15$ vertices and all blocks with whiskers where the block has at most $11$ vertices. This significantly extends previous computational results.
