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arXiv · 2212.05708

Cohen-Macaulay Property of Binomial Edge Ideals with Girth of Graphs

Kamalesh Saha·Indranath Sengupta
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Abstract

Conca and Varbaro (Invent. Math. 221 (2020), no. 3) showed the equality of depth of a graded ideal and its initial ideal in a polynomial ring when the initial ideal is square-free. In this paper, we give some beautiful applications of this fact in the study of Cohen-Macaulay binomial edge ideals. We prove that for the characterization of Cohen-Macaulay binomial edge ideals, it is enough to consider only "biconnected graphs with some whisker attached" and this done by investigating the initial ideals. We give several necessary conditions for a binomial edge ideal to be Cohen-Macaulay in terms of smaller graphs. Also, under a hypothesis, we give a sufficient condition for Cohen-Macaulayness of binomial edge ideals in terms of blocks of graphs. Moreover, we show that a graph with Cohen-Macaulay binomial edge ideal has girth less than $5$ or equal to infinity.

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Kamalesh Saha, Indranath Sengupta. 2022-12-12. Cohen-Macaulay Property of Binomial Edge Ideals with Girth of Graphs. https://arxiv.org/abs/2212.05708

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