Regularity of powers of edge ideals: from local properties to global bounds

Banerjee, Arindam; Beyarslan, Selvi Kara; Hà, Huy Tài

doi:10.5802/alco.119

Banerjee, Arindam ¹; Beyarslan, Selvi Kara ²; Hà, Huy Tài ³

¹ Ramakrishna Mission Vivekananda Educational and Research Institute Belur, West Bengal, India
² University of South Alabama Dept. of Mathematics and Statistics 411 University Boulevard North Mobile AL 36688-0002, USA
³ Tulane University Dept. of Mathematics 6823 St. Charles Ave. New Orleans LA 70118, USA

Algebraic Combinatorics, Volume 3 (2020) no. 4, pp. 839-854.

Abstract

Let $I = I (G)$ be the edge ideal of a graph $G$ . We give various general upper bounds for the regularity function $reg I^{s}$ , for $s \geq 1$ , addressing a conjecture made by the authors and Alilooee. When $G$ is a gap-free graph and locally of regularity 2, we show that $reg I^{s} = 2 s$ for all $s \geq 2$ . This is a weaker version of a conjecture of Nevo and Peeva. Our method is to investigate the regularity function $reg I^{s}$ , for $s \geq 1$ , via local information of $I$ .

Received: 2019-04-07
Revised: 2020-03-02
Accepted: 2020-03-18
Published online: 2020-08-20

DOI: 10.5802/alco.119
Classification: 05E40, 13A15, 13D02
Keywords: Castelnuovo–Mumford regularity, edge ideals, powers of ideals.
Author's affiliations:

Banerjee, Arindam ¹; Beyarslan, Selvi Kara ²; Hà, Huy Tài ³

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     title = {Regularity of powers of edge ideals: from local properties to global bounds},
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Banerjee, Arindam; Beyarslan, Selvi Kara; Hà, Huy Tài. Regularity of powers of edge ideals: from local properties to global bounds. Algebraic Combinatorics, Volume 3 (2020) no. 4, pp. 839-854. doi : 10.5802/alco.119. https://alco.centre-mersenne.org/articles/10.5802/alco.119/

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