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; Beyarslan, Selvi Kara; Hà, Huy Tài

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

@article{ALCO_2020__3_4_839_0,
     author = {Banerjee, Arindam and Beyarslan, Selvi Kara and H\`a, Huy T\`ai},
     title = {Regularity of powers of edge ideals: from local properties to global bounds},
     journal = {Algebraic Combinatorics},
     pages = {839--854},
     publisher = {MathOA foundation},
     volume = {3},
     number = {4},
     year = {2020},
     doi = {10.5802/alco.119},
     language = {en},
     url = {alco.centre-mersenne.org/item/ALCO_2020__3_4_839_0/}
}

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/item/ALCO_2020__3_4_839_0/

References

[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 | Article | MR 3713523 | Zbl 1372.05046

[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 | Article | MR 3983296 | Zbl 1417.05093

[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 | Article | MR 3228618 | Zbl 1307.13018

[4] Banerjee, Arindam The regularity of powers of edge ideals, J. Algebraic Combin., Volume 41 (2015) no. 2, pp. 303-321 | Article | MR 3306074 | Zbl 1318.13001

[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 | Article | MR 3922638 | Zbl 1418.13014

[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 | Article | MR 3417259 | Zbl 1369.13027

[7] Bruns, Winfried; Herzog, Jürgen Cohen–Macaulay rings, Cambridge university press, 1998 no. 39 | Zbl 0909.13005

[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 | Article | MR 2168694 | Zbl 1073.05054

[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 | Article | MR 3908353 | Zbl 07052595

[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 | Article | MR 1711319 | Zbl 0974.13015

[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 | Article | MR 3070118 | Zbl 1307.13021

[12] Eisenbud, David Commutative algebra. With a view toward algebraic geometry, Graduate Texts in Mathematics, Volume 150, Springer-Verlag, New York, 1995, xvi+785 pages | MR 1322960 | Zbl 0819.13001

[13] Erey, Nursel Powers of edge ideals with linear resolutions, Comm. Algebra, Volume 46 (2018) no. 9, pp. 4007-4020 | Article | MR 3820612 | Zbl 1428.13019

[14] Erey, Nursel Powers of ideals associated to $(C_{4}, 2 K_{2})$ -free graphs, J. Pure Appl. Algebra, Volume 223 (2019) no. 7, pp. 3071-3080 | Article | MR 3912959 | Zbl 1431.05154

[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 1171260 | Zbl 0741.13006

[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 | Article | MR 2375493 | Zbl 1147.05051

[17] Harary, Frank Graph theory, Addison-Wesley Publishing Co., Reading, Mass.-Menlo Park, Calif.-London, 1969, ix+274 pages | Zbl 0182.57702

[18] Herzog, Jürgen A generalization of the Taylor complex construction, Comm. Algebra, Volume 35 (2007) no. 5, pp. 1747-1756 | Article | MR 2317642 | Zbl 1121.13013

[19] Herzog, Jürgen; Hibi, Takayuki Monomial ideals, Graduate Texts in Mathematics, Volume 260, Springer-Verlag London, Ltd., London, 2011 | MR 2724673 | Zbl 1206.13001

[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 | Article | MR 2091479 | Zbl 1091.13013

[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 | Article | MR 3272076 | Zbl 1303.05218

[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 0441987 | Zbl 0351.13009

[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 | Article | MR 3757147 | Zbl 1381.05033

[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 | Article | MR 1621961 | Zbl 0929.13004

[27] Lyubeznik, Gennady The minimal non-Cohen–Macaulay monomial ideals, J. Pure Appl. Algebra, Volume 51 (1988) no. 3, pp. 261-266 | Article | MR 946577 | Zbl 0655.13020

[28] Miller, Ezra; Sturmfels, Bernd Combinatorial commutative algebra, Graduate Texts in Mathematics, Volume 227, Springer Science & Business Media, 2004 | Article | Zbl 1090.13001

[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 | Article | MR 3573376 | Zbl 1372.13012

[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 | Article | MR 2739498 | Zbl 1232.05092

[31] Nevo, Eran; Peeva, Irena $C_{4}$ -free edge ideals, J. Algebraic Combin., Volume 37 (2013) no. 2, pp. 243-248 | Article | MR 3011341 | Zbl 1274.13028

[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, Volume 41, Birkhäuser, 2007 | Zbl 1157.13302

[34] Sullivant, Seth Combinatorial symbolic powers, J. Algebra, Volume 319 (2008) no. 1, pp. 115-142 | Article | MR 2378064 | Zbl 1133.13027

[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 | Article | MR 2158746 | Zbl 1100.13024

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

[37] Wegner, Gerd $d$ -collapsing and nerves of families of convex sets, Arch. Math. (Basel), Volume 26 (1975), pp. 317-321 | Article | MR 375333 | Zbl 0308.52005

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