临界图中的复制与单项理想的持久性
Replication in critical graphs and the persistence of monomial ideals
摘要：
受关于多项式环中的无平方单项式理想的问题的启发，在2010年，Francisco等人证明了对于每个正整数k和每个k-临界（即，临界k-色）图，存在一个顶点集，其复制产生一个（k+1）-临界图. (The图的顶点集合W的复制是将W中的每个顶点w的副本添加到W中的操作，每次一个，并将其连接到w及其所有邻居。我们反驳的猜想提供了一个无限家庭的反例。此外，该家族的最小成员回答了Herzog和Hibi关于多项式环中无平方单项式理想的深度函数的一个问题，以及关于此类理想的持久性的一个相关问题。
英文摘要：
Motivated by questions about square-free monomial ideals in polynomial rings, in 2010 Francisco et al. conjectured that for every positive integer k and every k-critical (i.e., critically k-chromatic) graph, there is a set of vertices whose replication produces a (k+1)-critical graph. (The replication of a set W of vertices of a graph is the operation that adds a copy of each vertex w in W, one at a time, and connects it to w and all its neighbours.) We disprove the conjecture by providing an infinite family of counterexamples. Furthermore, the smallest member of the family answers a question of Herzog and Hibi concerning the depth functions of square-free monomial ideals in polynomial rings, and a related question on the persistence property of such ideals.
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