最大最小顶点覆盖和Betti表的大小
Max Min vertex cover and the size of Betti tables
摘要：
设$G$是$n$顶点上不含孤立点的有限单图，$I(G) \subseteq S = K[x_1, \dots, x_n]$是它的边理想.本文研究了度量投射维数的整数对和$S/I(G)$的正则性。我们证明了如果$S/I(G)$的投射维数达到最小值$2\sqrt{n}-2$，那么除了一个例外，它的正则性一定是1。当正则性达到最小值1时，我们也给出了$S/I(G)$的投射维数谱的完整描述。
英文摘要：
Let $G$ be a finite simple graph on $n$ vertices, that contains no isolated vertices, and let $I(G) \subseteq S = K[x_1, \dots, x_n]$ be its edge ideal. In this paper, we study the pair of integers that measure the projective dimension and the regularity of $S/I(G)$. We show that if the projective dimension of $S/I(G)$ attains its minimum value $2\sqrt{n}-2$ then, with only one exception, the its regularity must be 1. We also provide a full description for the spectrum of the projective dimension of $S/I(G)$ when the regularity attains its minimum value 1.
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