Advertisement

A Beginner’s Guide to Edge and Cover Ideals

  • Adam Van TuylEmail author
Chapter
Part of the Lecture Notes in Mathematics book series (LNM, volume 2083)

Abstract

Monomial ideals, although intrinsically interesting, play an important role in studying the connections between commutative algebra and combinatorics. Broadly speaking, problems in combinatorics are encoded into monomial ideals, which then allow us to use techniques and methods in commutative algebra to solve the original question. Stanley’s proof of the Upper Bound Conjecture [180] for simplicial spheres is seen as one of the early highlights of exploiting this connection between two fields. To bridge these two areas of mathematics, Stanley used square-free monomial ideals.

Keywords

Simplicial Complex Chromatic Number Commutative Algebra Betti Number Linear Resolution 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Notes

Acknowledgements

I would like to thank the organizers of MONICA, Anna M. Bigatti, Philippe Gimenez, and Eduardo Sáenz-de-Cabezón, for the invitation to participate in this conference. As well, I would like to thank all the participants for stimulating discussions and their feedback. I would also like to thank Ben Babcock, Ashwini Bhat, Jen Biermann, Chris Francisco, Tai Hà, Andrew Hoefel, and Ştefan Tohǎneanu for their feedback on preliminary drafts. The author was supported in part by an NSERC Discovery Grant.

References

  1. 2.
    A. Alilooee, S. Faridi, Betti numbers of path ideals of cycles and line (2011). Preprint [arXiv:1110.6653v1]Google Scholar
  2. 20.
    J. Biermann, Cellular structure on the minimal resolution of the edge ideal of the complement of the n-cycle (2011). PreprintGoogle Scholar
  3. 27.
    R. Bouchat, H.T. Hà, A. O’Keefe, Path ideals of rooted trees and their graded Betti numbers. J. Comb. Theory Ser. A 118, 2411–2425 (2011)zbMATHCrossRefGoogle Scholar
  4. 30.
    W. Bruns, J. Herzog, Cohen-Macaulay rings, revised edn. (Cambridge University Press, Cambridge, 1998)Google Scholar
  5. 33.
    J. Chen, S. Morey, A. Sung, The stable set of associated primes of the ideal of a graph. Rocky Mt. J. Math. 32, 71–89 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 34.
    R.-X. Chen, Minimal free resolutions of linear edge ideals. J. Algebra 324, 3591–3613 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 35.
    M. Chudnovsky, N. Robertson, P. Seymour, R. Thomas, The strong perfect graph theorem. Ann. Math. (2) 164, 51–229 (2006)Google Scholar
  8. 40.
    A. Conca, E. De Negri, M-Sequences, graph ideals and ladder ideals of linear type. J. Algebra 211, 599–624 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 43.
    A. Corso, U. Nagel, Specializations of Ferrers ideals. J. Algebr. Comb. 28, 425–437 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 44.
    A. Corso, U. Nagel, Monomial and toric ideals associated to Ferrers graphs. Trans. Am. Math. Soc. 361, 1371–1395 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 46.
    K. Dalili, M. Kummini, Dependence of Betti numbers on characteristic (2010) [arXiv:1009.4243]Google Scholar
  12. 48.
    A. Dochtermann, A. Engström, Algebraic properties of edge ideals via combinatorial topology. Electron. J. Combin. 16 (2009). Special volume in honor of Anders Bjorner, Research Paper 2, 24 pp.Google Scholar
  13. 49.
    A. Dochtermann, A. Engström, Cellular resolutions of cointerval ideals. Math. Z. 270, 145–163 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 54.
    D. Eisenbud, M. Green, K. Hulek, S. Popescu, Restricting linear syzygies: Algebra and geometry. Compos. Math. 141, 1460–1478 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 56.
    S. Eliahou, M. Kervaire, Minimal resolutions of some monomial ideals. J. Algebra 129, 1–25 (1990)MathSciNetzbMATHCrossRefGoogle Scholar
  16. 57.
    E. Emtander, A class of hypergraphs that generalizes chordal graphs. Math. Scand. 106, 50–66 (2010)MathSciNetzbMATHGoogle Scholar
  17. 58.
    E. Emtander, Betti numbers of hypergraphs. Commun. Algebra 37, 1545–1571 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  18. 59.
    S. Faridi, The facet ideal of a simplicial complex. Manuscripta Math. 109, 159–174 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  19. 60.
    G. Fatabbi, On the resolution of ideals of fat points. J. Algebra 242, 92–108 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  20. 61.
    O. Fernández-Ramos, P. Gimenez, First nonlinear syzygies of ideals associated to graphs. Commun. Algebra 37, 1921–1933 (2009)zbMATHCrossRefGoogle Scholar
  21. 64.
    C.A. Francisco, Resolutions of small sets of fat points. J. Pure Appl. Algebra 203, 220–236 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  22. 65.
    C. Francisco, H.T. Hà, A. Van Tuyl, Splittings of monomial ideals. Proc. Am. Math. Soc. 137, 3271–3282 (2009)zbMATHCrossRefGoogle Scholar
  23. 66.
    C.A. Francisco, H.T. Hà, A. Van Tuyl, A conjecture on critical graphs and connections to the persistence of associated primes. Discrete Math. 310, 2176–2182 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  24. 67.
    C.A. Francisco, H.T. Hà, A. Van Tuyl, Associated primes of monomial ideals and odd holes in graphs. J. Algebr. Comb. 32, 287–301 (2010)zbMATHCrossRefGoogle Scholar
  25. 68.
    C.A. Francisco, H.T. Hà, A. Van Tuyl, Colorings of hypergraphs, perfect graphs, and associated primes of powers of monomial ideals. J. Algebra 331, 224–242 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  26. 69.
    C.A. Francisco, A. Hoefel, A. Van Tuyl, EdgeIdeals: A package for (hyper)graphs. J. Softw. Algebra Geom. 1, 1–4 (2009)MathSciNetGoogle Scholar
  27. 70.
    R. Fröberg, On Stanley-Reisner rings, in Topics in Algebra, Part 2 (Warsaw, 1988). PWN, vol. 26 (Banach Center Publ., Warsaw, 1990), pp. 57–70Google Scholar
  28. 80.
    D. Grayson, M. Stillman, Macaulay2, A software system for research in algebraic geometry (2011). Available at: http://www.math.uiuc.edu/Macaulay2
  29. 83.
    H.T. Hà, A. Van Tuyl, Splittable ideals and the resolution of monomial ideals. J. Algebra 309, 405–425 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  30. 84.
    H.T. Hà, A. Van Tuyl, Resolutions of square-free monomial ideals via facet ideals: A survey. Contemp. Math. 448, 91–117 (2007)CrossRefGoogle Scholar
  31. 85.
    H.T. Hà, A. Van Tuyl, Monomial ideals, edge ideals of hypergraphs, and their graded Betti numbers. J. Algebr. Combin. 27, 215–245 (2008)zbMATHCrossRefGoogle Scholar
  32. 86.
    J. He, A. Van Tuyl, Algebraic properties of the path ideal of a tree. Commun. Algebra 38, 1725–1742 (2010)zbMATHCrossRefGoogle Scholar
  33. 91.
    J. Herzog, T. Hibi, in Monomial Ideals. GTM, vol. 260 (Springer, Berlin, 2010)Google Scholar
  34. 92.
    J. Herzog, T. Hibi, F. Hreinsdóttir, T. Kahle, J. Rauh, Binomial edge ideals and conditional independence statements. Adv. Appl. Math. 45, 317–333 (2010)zbMATHCrossRefGoogle Scholar
  35. 93.
    J. Herzog, T. Hibi, X. Zheng, Monomial ideals whose powers have a linear resolution. Math. Scand. 95, 23–32 (2004)MathSciNetzbMATHGoogle Scholar
  36. 105.
    T. Hibi, K. Kimura, S. Murai, Betti numbers of chordal graphs and f-vectors of simplicial complexes. J. Algebra 323, 1678–1689 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  37. 106.
    A. Hoefel, G. Whieldon, Linear quotients of the square of the edge ideal of the anticycle (2011). Preprint [arXiv:1106.2348v2]Google Scholar
  38. 107.
    N. Horwitz, Linear resolutions of quadratic monomial ideals. J. Algebra 318, 981–1001 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  39. 117.
    T. Kaiser, M. Stehlik, R. Skrekovski, Replication in critical graphs and the persistence of monomial ideals (2013). Preprint [arXiv:1301.6983.v2]Google Scholar
  40. 118.
    M. Katzman, Characteristic-independence of Betti numbers of graph ideals. J. Comb. Theory Ser. A 113, 435–454 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  41. 121.
    K. Kimura, Non-vanishingness of Betti numbers of edge ideals (2011). Preprint [arXiv:1110.2333v3]Google Scholar
  42. 125.
    M. Kummini, Regularity, depth and arithmetic rank of bipartite edge ideals. J. Algebr. Comb. 30, 429–445 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  43. 138.
    J. Martinez-Bernal, S. Morey, R. Villarreal, Associated primes of powers of edge ideals (2011). Preprint [arXiv:1103.0992v3]Google Scholar
  44. 141.
    E.  Miller, B.  Sturmfels, in Combinatorial Commutative Algebra. Graduate Texts in Mathematics, vol. 227 (Springer, New York, 2005)Google Scholar
  45. 143.
    S. Moradi, D. Kiani, Bounds for the regularity of edge ideal of vertex decomposable and shellable graphs. Bull. Iran. Math. Soc. 36, 267–277 (2010)MathSciNetzbMATHGoogle Scholar
  46. 144.
    S. Morey, E. Reyes, R. Villarreal, Cohen-Macaulay, shellable and unmixed clutters with a perfect matching of König type. J. Pure Appl. Algebra 212, 1770–1786 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  47. 145.
    S. Morey, R. Villarreal, Edge ideals: Algebraic and combinatorial properties, in Progress in Commutative Algebra, vol. 1 (de Gruyter, Berlin, 2012), pp. 85–126Google Scholar
  48. 150.
    E. Nevo, Regularity of edge ideals of C 4-free graphs via the topology of the lcm-lattice. J. Comb. Theory Ser. A 118, 491–501 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  49. 151.
    E. Nevo, I. Peeva, Linear resolutions of powers of edge ideals (2010). PreprintGoogle Scholar
  50. 156.
    I. Peeva, Graded Syzygies (Springer, New York, 2010)Google Scholar
  51. 169.
    E. Scheinerman, D. Ullman, Fractional Graph Theory. A Rational Approach to the Theory of Graphs (Wiley, New York, 1997)Google Scholar
  52. 173.
    R.Y. Sharp, Steps in Commutative Algebra, 2nd edn. (Cambridge University Press, Cambridge, 2000)zbMATHGoogle Scholar
  53. 175.
    A. Simis, W. Vasconcelos, R.H. Villarreal, On the ideal theory of graphs. J. Algebra 167, 389–416 (1994)MathSciNetzbMATHCrossRefGoogle Scholar
  54. 180.
    R. Stanley, The Upper Bound Conjecture and Cohen-Macaulay rings. Stud. Appl. Math. 54, 135–142 (1975)MathSciNetzbMATHGoogle Scholar
  55. 182.
    R.P. Stanley, Combinatorics and Commutative Algebra (Birkhäuser, Basel, 1983)zbMATHGoogle Scholar
  56. 185.
    B. Sturmfels, S. Sullivant, Combinatorial secant varieties. Pure Appl. Math. Q. 2(3), Part 1, 867–891 (2006)Google Scholar
  57. 189.
    G. Valla, Betti numbers of some monomial ideals. Proc. Am. Math. Soc. 133, 57–63 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  58. 190.
    A. Van Tuyl, Sequentially Cohen-Macaulay bipartite graphs: vertex decomposability and regularity. Arch. Math. (Basel) 93, 451–459 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  59. 192.
    R.H. Villarreal, Cohen-Macaulay graphs. Manuscripta Math. 66, 277–293 (1990)MathSciNetzbMATHCrossRefGoogle Scholar
  60. 193.
    R.H. Villarreal, in Monomial Algebras. Monographs and Textbooks in Pure and Applied Mathematics, vol. 238 (Marcel Dekker, New York, 2001)Google Scholar
  61. 196.
    G. Whieldon, Jump sequences of edge ideals (2010). Preprint [arXiv:1012.0108v1]Google Scholar
  62. 197.
    R. Woodroofe, Matchings, coverings, and Castelnuovo-Mumford regularity (2010). Preprint [arXiv:1009.2756]Google Scholar
  63. 198.
    R. Woodroofe, Chordal and sequentially Cohen-Macaulay clutters. Electron. J. Comb. 18, Paper 208 (2011)Google Scholar
  64. 203.
    X. Zheng, Resolutions of facet ideals. Commun. Algebra 32, 2301–2324 (2004)zbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Department of Mathematical SciencesLakehead UniversityThunder BayCanada

Personalised recommendations