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Collectanea Mathematica

, Volume 69, Issue 2, pp 249–262 | Cite as

Improved bounds for the regularity of edge ideals of graphs

  • S. A. Seyed Fakhari
  • S. Yassemi
Article

Abstract

Let G be a graph with n vertices, let \(S={\mathbb {K}}[x_1,\dots ,x_n]\) be the polynomial ring in n variables over a field \({\mathbb {K}}\) and let I(G) denote the edge ideal of G. For every collection \({\mathcal {H}}\) of connected graphs with \(K_2\in {\mathcal {H}}\), we introduce the notions of \({{\mathrm{ind-match}}}_{{\mathcal {H}}}(G)\) and \({{\mathrm{min-match}}}_{{\mathcal {H}}}(G)\). It will be proved that the inequalities \({{\mathrm{ind-match}}}_{\{K_2, C_5\}}(G)\le \mathrm{reg}(S/I(G))\le {{\mathrm{min-match}}}_{\{K_2, C_5\}}(G)\) are true. Moreover, we show that if G is a Cohen–Macaulay graph with girth at least five, then \(\mathrm{reg}(S/I(G))={{\mathrm{ind-match}}}_{\{K_2, C_5\}}(G)\). Furthermore, we prove that if G is a paw-free and doubly Cohen–Macaulay graph, then \(\mathrm{reg}(S/I(G))={{\mathrm{ind-match}}}_{\{K_2, C_5\}}(G)\) if and only if every connected component of G is either a complete graph or a 5-cycle graph. Among other results, we show that for every doubly Cohen–Macaulay simplicial complex, the equality \(\mathrm{reg}({\mathbb {K}}[\Delta ])=\mathrm{dim}({\mathbb {K}}[\Delta ])\) holds.

Keywords

Edge ideal Castelnuovo–Mumford regularity Girth Matching Paw-free graph 

Mathematics Subject Classification

Primary: 13D02 05E99 Secondary: 13F55 

Notes

Acknowledgements

The authors thank the referee for careful reading of the paper and for useful comments.

References

  1. 1.
    Athanasiadis, C.: Some combinatorial properties of flag simplicial pseudomanifolds and spheres. Ark. Mat. 49, 17–29 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Biermann, J., Van Tuyl, A.: Balanced vertex decomposable simplicial complexes and their h-vectors. Electron. J. Combin. 20(3), 15 (2013)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Biyikŏglu, T., Civan, Y.: Vertex decomposable graphs, codismantlability, Cohen–Macaulayness and Castelnuovo–Mumford regularity. Electron. J. Combin. 21, 17 (2014)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Bruns, W., Herzog, J.: Cohen–Macaulay Rings, Cambridge Studies in Advanced Mathematics, vol. 39. Cambridge University Press, Cambridge (1993)zbMATHGoogle Scholar
  5. 5.
    Dao, H., Huneke, C., Schweig, J.: Bounds on the regularity and projective dimension of ideals associated to graphs. J. Algebraic Combin. 38, 7–55 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Eisenbud, D.: Commutative Algebra with a View Toward Algebraic Geometry. Graduate Texts in Mathematics, vol. 150. Springer, New York (1995)zbMATHGoogle Scholar
  7. 7.
    Fröberg, R.: A note on the Stanley–Reisner ring of a join and of a suspension. Manuscr. Math. 60, 89–91 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Hà, H.T.: Regularity of squarefree monomial ideals. In: Copper, S.M., Sather-Wagstaff, S. (eds.) Connections Between Algebra, Combinatorics, and Geometry. Springer Proceedings in Mathematics Statistics, vol. 76, pp. 251–276. Springer, New York (2014)CrossRefGoogle Scholar
  9. 9.
    Hà, H.T., Van Tuyl, A.: Monomial ideals, edge ideals of hypergraphs, and their graded Betti numbers. J. Algebraic Combin. 27, 215–245 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Hà, H.T., Woodroofe, R.: Results on the regularity of square-free monomial ideals. Adv. Appl. Math. 58, 21–36 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Herzog, J.: A generalization of the Taylor complex construction. Commun. Algebra 35, 1747–1756 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Hibi, T., Higashitani, A., Kimura, K., Tsuchiya, A.: Dominating induced matchings of finite graphs and regularity of edge ideals. J. Algebraic Combin. 43, 173–198 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Hoang, D.T., Minh, N.C., Trung, T.N.: Cohen–Macaulay graphs with large girth. J. Algebra Appl. 14(7), 1550112 (2015). (16 pp) MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Hoang, D.T., Trung, T.N.: A characterization of triangle–free Gorenstein graphs and Cohen–Macaulayness of second powers of edge ideals. J. Algebraic Combin. 43, 325–338 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Kalai, G., Meshulam, R.: Intersections of Leray complexes and regularity of monomial ideals. J. Combin. Theory Ser. A 113, 1586–1592 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Katzman, M.: Characteristic-independence of Betti numbers of graph ideals. J. Combin. Theory Ser. A 113, 435–454 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Khosh-Ahang, F., Moradi, S.: Regularity and projective dimension of edge ideal of \(C_5\)-free vertex decomposable graphs. Proc. Am. Math. Soc. 142, 1567–1576 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Kummini, M.: Regularity, depth and arithmetic rank of bipartite edge ideals. J. Algebraic Combin. 30, 429–445 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Mahmoudi, M., Mousivand, A., Crupi, M., Rinaldo, G., Terai, N., Yassemid, S.: Vertex decomposability and regularity of very well-covered graphs. J. Pure Appl. Algebra 215, 2473–2480 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Peeva, I.: Graded Syzygies, Algebra and Applications, vol. 14. Springer, London (2011)CrossRefzbMATHGoogle Scholar
  21. 21.
    Van Tuyl, A.: Sequentially Cohen–Macaulay bipartite graphs: vertex decomposability and regularity. Arch. Math. (Basel) 93, 451–459 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Walker, J.W.: Topology and combinatorics of ordered sets. Ph.D. Thesis, MIT (1981)Google Scholar
  23. 23.
    Woodroofe, R.: Matchings, coverings, and Castelnuovo–Mumford regularity. J. Commut. Algebra 6, 287–304 (2014)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Universitat de Barcelona 2017

Authors and Affiliations

  1. 1.School of Mathematics, Statistics and Computer Science, College of ScienceUniversity of TehranTehranIran

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