• Mohammadi, Fatemeh (Institute of Science and Technology) ;
  • Moradi, Somayeh (Department of Mathematics Ilam University, School of Mathematics Institute for Research in Fundamental Sciences (IPM))
  • Received : 2014.06.18
  • Published : 2015.05.31


Let G be a graph on the vertex set $V(G)=\{x_1,{\cdots},x_n\}$ with the edge set E(G), and let $R=K[x_1,{\cdots},x_n]$ be the polynomial ring over a field K. Two monomial ideals are associated to G, the edge ideal I(G) generated by all monomials $x_i,x_j$ with $\{x_i,x_j\}{\in}E(G)$, and the vertex cover ideal $I_G$ generated by monomials ${\prod}_{x_i{\in}C}{^{x_i}}$ for all minimal vertex covers C of G. A minimal vertex cover of G is a subset $C{\subset}V(G)$ such that each edge has at least one vertex in C and no proper subset of C has the same property. Indeed, the vertex cover ideal of G is the Alexander dual of the edge ideal of G. In this paper, for an unmixed bipartite graph G we consider the lattice of vertex covers $L_G$ and we explicitly describe the minimal free resolution of the ideal associated to $L_G$ which is exactly the vertex cover ideal of G. Then we compute depth, projective dimension, regularity and extremal Betti numbers of R/I(G) in terms of the associated lattice.



Supported by : IPM


  1. D. Bayer, H. Charalambous, and S. Popescu, Extremal Betti numbers and applications to monomial ideals, J. Algebra 221 (1999), no. 2, 497-512.
  2. D. Bayer and M. Stillman, Macaulay, A system for computation in algebraic geometry and commutative algebra.
  3. H. T. Ha and A. Van Tuyl, Resolutions of square-free monomial ideals via facet ideals: a survey, Algebra, geometry and their interactions, 91-117, Contemp. Math., 448, Amer. Math. Soc., Providence, RI, 2007.
  4. H. T. Ha and A. Van Tuyl, Monomial ideals, edge ideals of hypergraphs, and their graded Betti numbers, J. Algebraic Combin. 27 (2008), no. 2, 215-245.
  5. J. Herzog and T. Hibi, Distributive lattices, bipartite graphs and Alexander duality, J. Algebraic Combin. 22 (2005), no. 3, 289-302.
  6. J. Herzog, T. Hibi, and H. Ohsugi, Unmixed bipartite graphs and sublattices of the Boolean lattices, J. Algebraic Combin. 30 (2009), no. 4, 415-420.
  7. J. Herzog, T. Hibi, and X. Zheng, Cohen-Macaulay chordal graphs, J. Combin. Theory Ser. A 113 (2006), no. 5, 911-916.
  8. J. Herzog, T. Hibi, and X. Zheng, The monomial ideal of a finite meet-semilattice, Trans. Amer. Math. Soc. 358 (2006), no. 9, 4119-4134.
  9. T. Hibi, Distributive lattices, affine semigroup rings and algebras with straightening laws, in Commutative algebra and combinatorics (Kyoto, 1985), 93-109, Adv. Stud. Pure Math., 11, North-Holland, Amsterdam, 1987.
  10. M. Katzman, Characteristic-independence of Betti numbers of graph ideals, J. Combin. Theory Ser. A 113 (2006), no. 3, 435-454.
  11. F. Khosh-Ahang and S. Moradi, Regularity and projective dimension of edge ideal of $C_5$-free vertex decomposable graphs, Proc. Amer. Math. Soc. 142 (2014), no. 5, 1567-1576.
  12. K. Kimura, Non-vanishingness of Betti numbers of edge ideals, Harmony of Grobner bases and the modern industrial society, 153-168, World Sci. Publ., Hackensack, NJ, 2012.
  13. M. Kummini, Regularity, depth and arithmetic rank of bipartite edge ideals, J. Algebraic Combin. 30 (2009), 429-445.
  14. N. Terai, Alexander duality theorem and Stanley-Reisner rings. Free resolutions of coordinate rings of projective varieties and related topics, (Japanese) (Kyoto, 1998), Surikaisekikenkyusho Kokyuroku (1999), no. 1078, 174-184.
  15. A. Van Tuyl, Sequentially Cohen-Macaulay bipartite graphs: vertex decomposability and regularity, Arch. Math. (Basel) 93 (2009), no. 5, 451-459.
  16. R. H. Villarreal, Unmixed bipartite graphs, Rev. Colombiana Mat. 41 (2007), no. 2, 393-395.
  17. X. Zheng, Resolutions of facet ideals, Comm. Algebra 32 (2004), no. 6, 2301-2324.

Cited by