# ALGEBRAIC COMBINATORICS

Bounds for the regularity of product of edge ideals
Algebraic Combinatorics, Volume 5 (2022) no. 5, pp. 1015-1032.

Let $I$ and $J$ be edge ideals in a polynomial ring $R=𝕂\left[{x}_{1},...,{x}_{n}\right]$ with $I\subseteq J$. In this paper, we obtain a general upper and lower bound for the Castelnuovo–Mumford regularity of $IJ$ in terms of certain invariants associated with $I$ and $J$. Using these results, we explicitly compute the regularity of $IJ$ for several classes of edge ideals. In particular, we compute the regularity of $IJ$ when $J$ has a linear resolution. Finally, we compute the precise expression for the regularity of ${J}_{1}{J}_{2}\cdots {J}_{d}$, $d\in \left\{3,4\right\}$, where ${J}_{1},...,{J}_{d}$ are edge ideals, ${J}_{1}\subseteq {J}_{2}\subseteq \cdots \subseteq {J}_{d}$ and ${J}_{d}$ is the edge ideal of a complete graph.

Revised:
Accepted:
Published online:
DOI: 10.5802/alco.234
Classification: 13D02,  05E45,  05C70
Keywords: Castelnuovo–Mumford regularity, product of edge ideals, linear resolution
Banerjee, Arindam 1; Das, Priya 2; Selvaraja, S 3

1 Department of Mathematics Indian Institute of Technology Kharagpur 721302 India
2 Department of Mathematics National Institute of Technology Calicut Kerala 673601 India
3 Chennai Mathematical Institute H1 SIPCOT IT Park Siruseri Kelambakkam Chennai India 603103
@article{ALCO_2022__5_5_1015_0,
author = {Banerjee, Arindam and Das, Priya and Selvaraja, S},
title = {Bounds for the regularity of product of edge ideals},
journal = {Algebraic Combinatorics},
pages = {1015--1032},
publisher = {The Combinatorics Consortium},
volume = {5},
number = {5},
year = {2022},
doi = {10.5802/alco.234},
language = {en},
url = {https://alco.centre-mersenne.org/articles/10.5802/alco.234/}
}
TY  - JOUR
AU  - Banerjee, Arindam
AU  - Das, Priya
AU  - Selvaraja, S
TI  - Bounds for the regularity of product of edge ideals
JO  - Algebraic Combinatorics
PY  - 2022
DA  - 2022///
SP  - 1015
EP  - 1032
VL  - 5
IS  - 5
PB  - The Combinatorics Consortium
UR  - https://alco.centre-mersenne.org/articles/10.5802/alco.234/
UR  - https://doi.org/10.5802/alco.234
DO  - 10.5802/alco.234
LA  - en
ID  - ALCO_2022__5_5_1015_0
ER  - 
%0 Journal Article
%A Banerjee, Arindam
%A Das, Priya
%A Selvaraja, S
%T Bounds for the regularity of product of edge ideals
%J Algebraic Combinatorics
%D 2022
%P 1015-1032
%V 5
%N 5
%I The Combinatorics Consortium
%U https://doi.org/10.5802/alco.234
%R 10.5802/alco.234
%G en
%F ALCO_2022__5_5_1015_0
Banerjee, Arindam; Das, Priya; Selvaraja, S. Bounds for the regularity of product of edge ideals. Algebraic Combinatorics, Volume 5 (2022) no. 5, pp. 1015-1032. doi : 10.5802/alco.234. https://alco.centre-mersenne.org/articles/10.5802/alco.234/

[1] Banerjee, Arindam The regularity of powers of edge ideals, J. Algebraic Combin., Volume 41 (2015) no. 2, pp. 303-321 | DOI | MR | Zbl

[2] Banerjee, Arindam; Beyarslan, Selvi Kara; Huy Tài, Hà Regularity of edge ideals and their powers, Advances in algebra (Springer Proc. Math. Stat.), Volume 277, Springer, Cham, 2019, pp. 17-52 | DOI | MR | Zbl

[3] Benzaken, Claude; Crama, Yves; Duchet, Pierre; Hammer, Peter L.; Maffray, Frédéric More characterizations of triangulated graphs, J. Graph Theory, Volume 14 (1990) no. 4, pp. 413-422 | DOI | MR | Zbl

[4] Beyarslan, Selvi; Hà, Huy Tài; Trung, Trân Nam Regularity of powers of forests and cycles, J. Algebraic Combin., Volume 42 (2015) no. 4, pp. 1077-1095 | DOI | MR | Zbl

[5] Caviglia, Giulio Bounds on the Castelnuovo-Mumford regularity of tensor products, Proc. Amer. Math. Soc., Volume 135 (2007) no. 7, pp. 1949-1957 | DOI | MR | Zbl

[6] Chardin, Marc; Minh, Nguyen Cong; Trung, Ngo Viet On the regularity of products and intersections of complete intersections, Proc. Amer. Math. Soc., Volume 135 (2007) no. 6, pp. 1597-1606 | DOI | MR | Zbl

[7] Cimpoeaş, Mircea Some remarks on Borel type ideals, Comm. Algebra, Volume 37 (2009) no. 2, pp. 724-727 | DOI | MR | Zbl

[8] Conca, Aldo; Herzog, Jürgen Castelnuovo-Mumford regularity of products of ideals, Collect. Math., Volume 54 (2003) no. 2, pp. 137-152 | MR | Zbl

[9] Eisenbud, David; Huneke, Craig; Ulrich, Bernd The regularity of Tor and graded Betti numbers, Amer. J. Math., Volume 128 (2006) no. 3, pp. 573-605 | DOI | MR | Zbl

[10] Erey, Nursel Powers of edge ideals with linear resolutions, Comm. Algebra, Volume 46 (2018) no. 9, pp. 4007-4020 | DOI | MR | Zbl

[11] Erey, Nursel Powers of ideals associated to $\left({C}_{4},2{K}_{2}\right)$-free graphs, J. Pure Appl. Algebra, Volume 223 (2019) no. 7, pp. 3071-3080 | DOI | MR | Zbl

[12] Fröberg, Ralf On Stanley-Reisner rings, Topics in algebra, Part 2 (Warsaw, 1988) (Banach Center Publ.), Volume 26, PWN, Warsaw, 1990, pp. 57-70 | MR | Zbl

[13] Grayson, Daniel R.; Stillman, Michael E. Macaulay2, a software system for research in algebraic geometry, Available at http://www.math.uiuc.edu/Macaulay2/

[14] Herzog, Jürgen; Hibi, Takayuki Monomial ideals, Graduate Texts in Mathematics, 260, Springer-Verlag London, Ltd., London, 2011, xvi+305 pages | DOI | MR

[15] Jayanthan, A. V.; Narayanan, N.; Selvaraja, S. Regularity of powers of bipartite graphs, J. Algebraic Combin., Volume 47 (2018) no. 1, pp. 17-38 | DOI | MR | Zbl

[16] Jayanthan, A. V.; Selvaraja, S. Upper bounds for the regularity of powers of edge ideals of graphs, J. Algebra, Volume 574 (2021), pp. 184-205 | DOI | MR | Zbl

[17] Kalai, Gil; Meshulam, Roy Intersections of Leray complexes and regularity of monomial ideals, J. Combin. Theory Ser. A, Volume 113 (2006) no. 7, pp. 1586-1592 | DOI | MR | Zbl

[18] Katzman, Mordechai Characteristic-independence of Betti numbers of graph ideals, J. Combin. Theory Ser. A, Volume 113 (2006) no. 3, pp. 435-454 | DOI | MR | Zbl

[19] Miller, Ezra; Sturmfels, Bernd Combinatorial commutative algebra, Graduate Texts in Mathematics, 227, Springer-Verlag, New York, 2005, xiv+417 pages | MR

[20] Moradi, Somayeh; Kiani, Dariush Bounds for the regularity of edge ideal of vertex decomposable and shellable graphs, Bull. Iranian Math. Soc., Volume 36 (2010) no. 2, p. 267-277, 302 | MR | Zbl

[21] Selvaraja, S. Regularity of powers of edge ideals of product of graphs, J. Algebra Appl., Volume 17 (2018) no. 6, p. 1850128 | DOI | MR | Zbl

[22] Sidman, Jessica On the Castelnuovo-Mumford regularity of products of ideal sheaves, Adv. Geom., Volume 2 (2002) no. 3, pp. 219-229 | DOI | MR | Zbl

[23] Sturmfels, Bernd Four counterexamples in combinatorial algebraic geometry, J. Algebra, Volume 230 (2000) no. 1, pp. 282-294 | DOI | MR | Zbl

[24] Woodroofe, Russ Matchings, coverings, and Castelnuovo-Mumford regularity, J. Commut. Algebra, Volume 6 (2014) no. 2, pp. 287-304 | MR | Zbl

Cited by Sources: