Regularity of powers of edge ideals: from local properties to global bounds
Algebraic Combinatorics, Volume 3 (2020) no. 4, pp. 839-854.

Let I=I(G) be the edge ideal of a graph G. We give various general upper bounds for the regularity function regI s , for s1, addressing a conjecture made by the authors and Alilooee. When G is a gap-free graph and locally of regularity 2, we show that regI s =2s for all s2. This is a weaker version of a conjecture of Nevo and Peeva. Our method is to investigate the regularity function regI s , for s1, via local information of I.

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

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
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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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/

[1] Alilooee, Ali; Banerjee, Arindam Powers of edge ideals of regularity three bipartite graphs, J. Commut. Algebra, Volume 9 (2017) no. 4, pp. 441-454 | DOI | MR | Zbl

[2] Alilooee, Ali; Beyarslan, Selvi Kara; Selvaraja, S. Regularity of powers of edge ideals of unicyclic graphs, Rocky Mountain J. Math., Volume 49 (2019) no. 3, pp. 699-728 | DOI | MR | Zbl

[3] Bagheri, Amir; Chardin, Marc; Hà, Huy Tài The eventual shape of Betti tables of powers of ideals, Math. Res. Lett., Volume 20 (2013) no. 6, pp. 1033-1046 | DOI | MR | Zbl

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

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

[6] 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

[7] Bruns, Winfried; Herzog, Jürgen Cohen–Macaulay rings, Cambridge university press, 1998 no. 39 | Zbl

[8] Cameron, Kathie; Walker, Tracy The graphs with maximum induced matching and maximum matching the same size, Discrete Math., Volume 299 (2005) no. 1-3, pp. 49-55 | DOI | MR | Zbl

[9] Caviglia, Giulio; Hà, Huy Tài; Herzog, Jürgen; Kummini, Manoj; Terai, Naoki; Trung, Ngô Viêt Depth and regularity modulo a principal ideal, J. Algebraic Combin., Volume 49 (2019) no. 1, pp. 1-20 | DOI | MR | Zbl

[10] Cutkosky, S. Dale; Herzog, Jürgen; Trung, Ngô Viêt Asymptotic behaviour of the Castelnuovo–Mumford regularity, Compositio Math., Volume 118 (1999) no. 3, pp. 243-261 | DOI | MR | Zbl

[11] Dao, Hailong; Huneke, Craig; Schweig, Jay Bounds on the regularity and projective dimension of ideals associated to graphs, J. Algebraic Combin., Volume 38 (2013) no. 1, pp. 37-55 | DOI | MR | Zbl

[12] Eisenbud, David Commutative algebra. With a view toward algebraic geometry, Graduate Texts in Mathematics, 150, Springer-Verlag, New York, 1995, xvi+785 pages | MR | Zbl

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

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

[15] 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

[16] Hà, Huy Tài; Van Tuyl, Adam Monomial ideals, edge ideals of hypergraphs, and their graded Betti numbers, J. Algebraic Combin., Volume 27 (2008) no. 2, pp. 215-245 | DOI | MR | Zbl

[17] Harary, Frank Graph theory, Addison-Wesley Publishing Co., Reading, Mass.-Menlo Park, Calif.-London, 1969, ix+274 pages | Zbl

[18] Herzog, Jürgen A generalization of the Taylor complex construction, Comm. Algebra, Volume 35 (2007) no. 5, pp. 1747-1756 | DOI | MR | Zbl

[19] Herzog, Jürgen; Hibi, Takayuki Monomial ideals, Graduate Texts in Mathematics, 260, Springer-Verlag London, Ltd., London, 2011 | MR | Zbl

[20] Herzog, Jürgen; Hibi, Takayuki; Zheng, Xinxian Monomial ideals whose powers have a linear resolution, Math. Scand., Volume 95 (2004) no. 1, pp. 23-32 | DOI | MR | Zbl

[21] Hibi, Takayuki; Higashitani, Akihiro; Kimura, Kyouko; O’Keefe, Augustine B. Algebraic study on Cameron–Walker graphs, J. Algebra, Volume 422 (2015), pp. 257-269 | DOI | MR | Zbl

[22] Hochster, Melvin Cohen–Macaulay rings, combinatorics, and simplicial complexes, Ring theory, II (Proc. Second Conf., Univ. Oklahoma, Norman, Okla., 1975) (Lecture Notes in Pure and Appl. Math.), Volume 26 (1977), pp. 171-223 | MR | Zbl

[23] 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

[24] Jayanthan, A. V.; Selvaraja, S. Asymptotic behavior of Castelnuovo–Mumford regularity of edge ideals of very well-covered graphs (2017) (To appear in J. Comm. Algebra, https://arxiv.org/abs/1708.06883)

[25] Jayanthan, A. V.; Selvaraja, S. Upper bounds for the regularity of powers of edge ideals of graphs (2018) (https://arxiv.org/abs/1805.01412)

[26] Kodiyalam, Vijay Asymptotic behaviour of Castelnuovo–Mumford regularity, Proc. Amer. Math. Soc., Volume 128 (2000) no. 2, pp. 407-411 | DOI | MR | Zbl

[27] Lyubeznik, Gennady The minimal non-Cohen–Macaulay monomial ideals, J. Pure Appl. Algebra, Volume 51 (1988) no. 3, pp. 261-266 | DOI | MR | Zbl

[28] Miller, Ezra; Sturmfels, Bernd Combinatorial commutative algebra, Graduate Texts in Mathematics, 227, Springer Science & Business Media, 2004 | DOI | Zbl

[29] Moghimian, Mahdiyeh; Fakhari, Seyed Amin Seyed; Yassemi, Siamak Regularity of powers of edge ideal of whiskered cycles, Comm. Algebra, Volume 45 (2017) no. 3, pp. 1246-1259 | DOI | MR | Zbl

[30] Nevo, Eran Regularity of edge ideals of C 4 -free graphs via the topology of the lcm-lattice, J. Combin. Theory Ser. A, Volume 118 (2011) no. 2, pp. 491-501 | DOI | MR | Zbl

[31] Nevo, Eran; Peeva, Irena C 4 -free edge ideals, J. Algebraic Combin., Volume 37 (2013) no. 2, pp. 243-248 | DOI | MR | Zbl

[32] Norouzi, Pooran; Fakhari, Seyed Amin Seyed; Yassemi, Siamak Regularity of Powers of edge ideal of very well-covered graphs (2017) (https://arxiv.org/abs/1707.04874)

[33] Stanley, Richard P. Combinatorics and commutative algebra, Progress in Mathematics, 41, Birkhäuser, 2007 | Zbl

[34] Sullivant, Seth Combinatorial symbolic powers, J. Algebra, Volume 319 (2008) no. 1, pp. 115-142 | DOI | MR | Zbl

[35] Trung, Ngô Viêt; Wang, Hsin-Ju On the asymptotic linearity of Castelnuovo–Mumford regularity, J. Pure Appl. Algebra, Volume 201 (2005) no. 1-3, pp. 42-48 | DOI | MR | Zbl

[36] Villarreal, Rafael Monomial algebras, CRC Press, 2018 | DOI

[37] Wegner, Gerd d-collapsing and nerves of families of convex sets, Arch. Math. (Basel), Volume 26 (1975), pp. 317-321 | DOI | MR | Zbl

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