## Associated primes of powers of edge ideals and ear decompositions of graphs

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- by Ha Minh Lam and Ngo Viet Trung PDF
- Trans. Amer. Math. Soc.
**372**(2019), 3211-3236 Request permission

## Abstract:

In this paper, we give a complete description of the associated primes of every power of the edge ideal in terms of generalized ear decompositions of the graph. This result establishes a surprising relationship between two seemingly unrelated notions of commutative algebra and combinatorics. It covers all previous major results in this topic and has several interesting consequences.## References

- Arindam Banerjee,
*The regularity of powers of edge ideals*, J. Algebraic Combin.**41**(2015), no. 2, 303–321. MR**3306074**, DOI 10.1007/s10801-014-0537-2 - M. Brodmann,
*Asymptotic stability of $\textrm {Ass}(M/I^{n}M)$*, Proc. Amer. Math. Soc.**74**(1979), no. 1, 16–18. MR**521865**, DOI 10.1090/S0002-9939-1979-0521865-8 - M. Brodmann,
*The asymptotic nature of the analytic spread*, Math. Proc. Cambridge Philos. Soc.**86**(1979), no. 1, 35–39. MR**530808**, DOI 10.1017/S030500410000061X - Winfried Bruns and Jürgen Herzog,
*Cohen-Macaulay rings*, Cambridge Studies in Advanced Mathematics, vol. 39, Cambridge University Press, Cambridge, 1993. MR**1251956** - Janet Chen, Susan Morey, and Anne Sung,
*The stable set of associated primes of the ideal of a graph*, Rocky Mountain J. Math.**32**(2002), no. 1, 71–89. MR**1911348**, DOI 10.1216/rmjm/1030539608 - S. Dale Cutkosky, Jürgen Herzog, and Ngô Viêt Trung,
*Asymptotic behaviour of the Castelnuovo-Mumford regularity*, Compositio Math.**118**(1999), no. 3, 243–261. MR**1711319**, DOI 10.1023/A:1001559912258 - N. T. Dung, N. T. T. Tam, H. L. Truong, and H. N. Yen,
*Critical paired dominating sets and irreducible decompositions of powers of edge ideals*, Acta Math. Vietnam. (2018). https://doi.org/10.1007/s40306-018-0273-0 - David Eisenbud,
*Commutative algebra*, Graduate Texts in Mathematics, vol. 150, Springer-Verlag, New York, 1995. With a view toward algebraic geometry. MR**1322960**, DOI 10.1007/978-1-4612-5350-1 - David Eisenbud and Joe Harris,
*Powers of ideals and fibers of morphisms*, Math. Res. Lett.**17**(2010), no. 2, 267–273. MR**2644374**, DOI 10.4310/MRL.2010.v17.n2.a6 - David Eisenbud and Bernd Ulrich,
*Notes on regularity stabilization*, Proc. Amer. Math. Soc.**140**(2012), no. 4, 1221–1232. MR**2869107**, DOI 10.1090/S0002-9939-2011-11270-X - Christopher A. Francisco, Huy Tài Hà, and Adam Van Tuyl,
*Colorings of hypergraphs, perfect graphs, and associated primes of powers of monomial ideals*, J. Algebra**331**(2011), 224–242. MR**2774655**, DOI 10.1016/j.jalgebra.2010.10.025 - András Frank,
*Conservative weightings and ear-decompositions of graphs*, Combinatorica**13**(1993), no. 1, 65–81. MR**1221177**, DOI 10.1007/BF01202790 - T. Gallai,
*Neuer Beweis eines Tutte’schen Satzes*, Magyar Tud. Akad. Mat. Kutató Int. Közl.**8**(1963), 135–139 (German, with Russian summary). MR**166777** - Huy Tài Hà and Susan Morey,
*Embedded associated primes of powers of square-free monomial ideals*, J. Pure Appl. Algebra**214**(2010), no. 4, 301–308. MR**2558739**, DOI 10.1016/j.jpaa.2009.05.002 - Huy Tài Hà and Mengyao Sun,
*Squarefree monomial ideals that fail the persistence property and non-increasing depth*, Acta Math. Vietnam.**40**(2015), no. 1, 125–137. MR**3331937**, DOI 10.1007/s40306-014-0104-x - Jürgen Herzog and Takayuki Hibi,
*The depth of powers of an ideal*, J. Algebra**291**(2005), no. 2, 534–550. MR**2163482**, DOI 10.1016/j.jalgebra.2005.04.007 - Jürgen Herzog and Takayuki Hibi,
*Monomial ideals*, Graduate Texts in Mathematics, vol. 260, Springer-Verlag London, Ltd., London, 2011. MR**2724673**, DOI 10.1007/978-0-85729-106-6 - Jürgen Herzog and Takayuki Hibi,
*Bounding the socles of powers of squarefree monomial ideals*, Commutative algebra and noncommutative algebraic geometry. Vol. II, Math. Sci. Res. Inst. Publ., vol. 68, Cambridge Univ. Press, New York, 2015, pp. 223–229. MR**3496867** - Jürgen Herzog, Takayuki Hibi, and Ngô Viêt Trung,
*Symbolic powers of monomial ideals and vertex cover algebras*, Adv. Math.**210**(2007), no. 1, 304–322. MR**2298826**, DOI 10.1016/j.aim.2006.06.007 - Jürgen Herzog and Ayesha Asloob Qureshi,
*Persistence and stability properties of powers of ideals*, J. Pure Appl. Algebra**219**(2015), no. 3, 530–542. MR**3279372**, DOI 10.1016/j.jpaa.2014.05.011 - Ha Thi Thu Hien and Ha Minh Lam,
*Combinatorial characterizations of the saturation and the associated primes of the fourth power of edge ideals*, Acta Math. Vietnam.**40**(2015), no. 3, 511–526. MR**3395652**, DOI 10.1007/s40306-015-0140-1 - Ha Thi Thu Hien, Ha Minh Lam, and Ngo Viet Trung,
*Saturation and associated primes of powers of edge ideals*, J. Algebra**439**(2015), 225–244. MR**3373370**, DOI 10.1016/j.jalgebra.2015.05.003 - Lê Tuân Hoa,
*Stability of associated primes of monomial ideals*, Vietnam J. Math.**34**(2006), no. 4, 473–487. MR**2286753** - Vijay Kodiyalam,
*Asymptotic behaviour of Castelnuovo-Mumford regularity*, Proc. Amer. Math. Soc.**128**(2000), no. 2, 407–411. MR**1621961**, DOI 10.1090/S0002-9939-99-05020-0 - L. Lovász,
*A note on factor-critical graphs*, Studia Sci. Math. Hungar.**7**(1972), 279–280. MR**335371** - José Martínez-Bernal, Susan Morey, and Rafael H. Villarreal,
*Associated primes of powers of edge ideals*, Collect. Math.**63**(2012), no. 3, 361–374. MR**2957976**, DOI 10.1007/s13348-011-0045-9 - Susan Morey,
*Depths of powers of the edge ideal of a tree*, Comm. Algebra**38**(2010), no. 11, 4042–4055. MR**2764849**, DOI 10.1080/00927870903286900 - Susan Morey and Rafael H. Villarreal,
*Edge ideals: algebraic and combinatorial properties*, Progress in commutative algebra 1, de Gruyter, Berlin, 2012, pp. 85–126. MR**2932582** - Giancarlo Rinaldo, Naoki Terai, and Ken-ichi Yoshida,
*Cohen-Macaulayness for symbolic power ideals of edge ideals*, J. Algebra**347**(2011), 1–22. MR**2846393**, DOI 10.1016/j.jalgebra.2011.09.007 - H. E. Robbins,
*Questions, Discussions, and Notes: A Theorem on Graphs, with an Application to a Problem of Traffic Control*, Amer. Math. Monthly**46**(1939), no. 5, 281–283. MR**1524589**, DOI 10.2307/2303897 - Rodney Y. Sharp,
*Convergence of sequences of sets of associated primes*, Proc. Amer. Math. Soc.**131**(2003), no. 10, 3009–3017. MR**1993206**, DOI 10.1090/S0002-9939-03-07038-2 - Aron Simis, Wolmer V. Vasconcelos, and Rafael H. Villarreal,
*On the ideal theory of graphs*, J. Algebra**167**(1994), no. 2, 389–416. MR**1283294**, DOI 10.1006/jabr.1994.1192 - Patrick Solé and Thomas Zaslavsky,
*The covering radius of the cycle code of a graph*, Discrete Appl. Math.**45**(1993), no. 1, 63–70. MR**1230281**, DOI 10.1016/0166-218X(93)90140-J - Yukihide Takayama,
*Combinatorial characterizations of generalized Cohen-Macaulay monomial ideals*, Bull. Math. Soc. Sci. Math. Roumanie (N.S.)**48(96)**(2005), no. 3, 327–344. MR**2165349** - Naoki Terai and Ngo Viet Trung,
*On the associated primes and the depth of the second power of squarefree monomial ideals*, J. Pure Appl. Algebra**218**(2014), no. 6, 1117–1129. MR**3153618**, DOI 10.1016/j.jpaa.2013.11.008 - Tran Nam Trung,
*Stability of depths of powers of edge ideals*, J. Algebra**452**(2016), 157–187. MR**3461061**, DOI 10.1016/j.jalgebra.2016.01.009 - Douglas B. West,
*Introduction to graph theory*, Prentice Hall, Inc., Upper Saddle River, NJ, 1996. MR**1367739**

## Additional Information

**Ha Minh Lam**- Affiliation: Institute of Mathematics, Vietnam Academy of Science and Technology, 18 Hoang Quoc Viet, 10307 Hanoi, Vietnam
- MR Author ID: 787483
- Email: hmlam@math.ac.vn
**Ngo Viet Trung**- Affiliation: International Centre for Research and Postgraduate Training, Institute of Mathematics, Vietnam Academy of Science and Technology,18 Hoang Quoc Viet, 10307 Hanoi, Vietnam
- MR Author ID: 207806
- Email: nvtrung@math.ac.vn
- Received by editor(s): January 3, 2018
- Received by editor(s) in revised form: July 12, 2018
- Published electronically: April 25, 2019
- Additional Notes: This work was supported by Vietnam National Foundation for Science and Technology Development under grant number 101.04-2017.19 and by Project ICRTM 01_2019.02 of the International Center for Research and Postgraduate Training in Mathematics, VAST
- © Copyright 2019 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**372**(2019), 3211-3236 - MSC (2010): Primary 13C05, 05C70, 05E40
- DOI: https://doi.org/10.1090/tran/7662
- MathSciNet review: 3988608