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arXiv · 2002.02523

Max Min vertex cover and the size of Betti tables

Huy Tai Ha·Takayuki Hibi
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Abstract

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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Huy Tai Ha, Takayuki Hibi. 2020-04-15. Max Min vertex cover and the size of Betti tables. https://arxiv.org/abs/2002.02523

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