Graph connectivity and binomial edge ideals

Banerjee, Arindam; Núñez-Betancourt, Luis

doi:10.1090/proc/13241

Graph connectivity and binomial edge ideals
HTML articles powered by AMS MathViewer

by Arindam Banerjee and Luis Núñez-Betancourt PDF

Proc. Amer. Math. Soc. 145 (2017), 487-499 Request permission

Abstract:

We relate homological properties of a binomial edge ideal $\mathcal {J}_G$ to invariants that measure the connectivity of a simple graph $G$. Specifically, we show if $R/\mathcal {J}_G$ is a Cohen-Macaulay ring, then graph toughness of $G$ is exactly $\frac {1}{2}$. We also give an inequality between the depth of $R/\mathcal {J}_G$ and the vertex-connectivity of $G$. In addition, we study the Hilbert-Samuel multiplicity and the Hilbert-Kunz multiplicity of $R/\mathcal {J}_G$.

References

Douglas Bauer, Hajo Broersma, and Edward Schmeichel, Toughness in graphs—a survey, Graphs Combin. 22 (2006), no. 1, 1–35. MR 2221006, DOI 10.1007/s00373-006-0649-0
Béla Bollobás, Modern graph theory, Graduate Texts in Mathematics, vol. 184, Springer-Verlag, New York, 1998. MR 1633290, DOI 10.1007/978-1-4612-0619-4
Holger Brenner, Irrational Hilbert-Kunz multiplicities, Preprint, arXiv:1305.5873 [math.AG], 2013.
M. P. Brodmann and R. Y. Sharp, Local cohomology: an algebraic introduction with geometric applications, Cambridge Studies in Advanced Mathematics, vol. 60, Cambridge University Press, Cambridge, 1998. MR 1613627, DOI 10.1017/CBO9780511629204
Winfried Bruns and Udo Vetter, Determinantal rings, Lecture Notes in Mathematics, vol. 1327, Springer-Verlag, Berlin, 1988. MR 953963, DOI 10.1007/BFb0080378
V. Chvátal, Tough graphs and Hamiltonian circuits, Discrete Math. 5 (1973), 215–228. MR 316301, DOI 10.1016/0012-365X(73)90138-6
Viviana Ene, Jürgen Herzog, and Takayuki Hibi, Cohen-Macaulay binomial edge ideals, Nagoya Math. J. 204 (2011), 57–68. MR 2863365, DOI 10.1215/00277630-1431831
Kazufumi Eto, Multiplicity and Hilbert-Kunz multiplicity of monoid rings, Tokyo J. Math. 25 (2002), no. 2, 241–245. MR 1948662, DOI 10.3836/tjm/1244208851
Kazufumi Eto and Ken-ichi Yoshida, Notes on Hilbert-Kunz multiplicity of Rees algebras, Comm. Algebra 31 (2003), no. 12, 5943–5976. MR 2014910, DOI 10.1081/AGB-120024861
Ralf Fröberg, On Stanley-Reisner rings, Topics in algebra, Part 2 (Warsaw, 1988) Banach Center Publ., vol. 26, PWN, Warsaw, 1990, pp. 57–70. MR 1171260
Daniel R. Grayson and Michael E. Stillman, Macaulay2, a software system for research in algebraic geometry, available at http://www.math.uiuc.edu/Macaulay2/.
Melvin Hochster and Craig Huneke, Tight closure in equal characteristic zero, Unpublished manuscript, available at http://www.math.lsa.umich.edu/~hochster/tcz.ps.
Melvin Hochster and Craig Huneke, Indecomposable canonical modules and connectedness, Commutative algebra: syzygies, multiplicities, and birational algebra (South Hadley, MA, 1992) Contemp. Math., vol. 159, Amer. Math. Soc., Providence, RI, 1994, pp. 197–208. MR 1266184, DOI 10.1090/conm/159/01509
Jürgen Herzog and Takayuki Hibi, Distributive lattices, bipartite graphs and Alexander duality, J. Algebraic Combin. 22 (2005), no. 3, 289–302. MR 2181367, DOI 10.1007/s10801-005-4528-1
Jürgen Herzog, Takayuki Hibi, Freyja Hreinsdóttir, Thomas Kahle, and Johannes Rauh, Binomial edge ideals and conditional independence statements, Adv. in Appl. Math. 45 (2010), no. 3, 317–333. MR 2669070, DOI 10.1016/j.aam.2010.01.003
Kazunori Matsuda and Satoshi Murai, Regularity bounds for binomial edge ideals, J. Commut. Algebra 5 (2013), no. 1, 141–149. MR 3084125, DOI 10.1216/jca-2013-5-1-141
P. Monsky, The Hilbert-Kunz function, Math. Ann. 263 (1983), no. 1, 43–49. MR 697329, DOI 10.1007/BF01457082
Masahiro Ohtani, Graphs and ideals generated by some 2-minors, Comm. Algebra 39 (2011), no. 3, 905–917. MR 2782571, DOI 10.1080/00927870903527584
C. Peskine and L. Szpiro, Dimension projective finie et cohomologie locale. Applications à la démonstration de conjectures de M. Auslander, H. Bass et A. Grothendieck, Inst. Hautes Études Sci. Publ. Math. 42 (1973), 47–119 (French). MR 374130
Giancarlo Rinaldo, Cohen-Macauley binomial edge ideals of small deviation, Bull. Math. Soc. Sci. Math. Roumanie (N.S.) 56(104) (2013), no. 4, 497–503. MR 3157095
Peter Schenzel and Sohail Zafar, Algebraic properties of the binomial edge ideal of a complete bipartite graph, An. Ştiinţ. Univ. “Ovidius” Constanţa Ser. Mat. 22 (2014), no. 2, 217–237. MR 3195706, DOI 10.2478/auom-2014-0043
Sohail Zafar and Zohaib Zahid, On the Betti numbers of some classes of binomial edge ideals, Electron. J. Combin. 20 (2013), no. 4, Paper 37, 14. MR 3158276, DOI 10.37236/3689