MAX MIN Vertex Cover and the Size of Betti Tables

Abstract

Let G be a finite simple graph on n vertices, that contains no isolated vertices, and let \(I(G) \subseteq S = K[x_1, \dots , x_n]\) be its edge ideal. In this paper, we study the pair of integers that measure the projective dimension and the regularity of S/I(G). We show that if \({{\,\mathrm{pd}\,}}(S/I(G))\) attains its minimum possible value \(2\sqrt{n}-2\) then, with only one exception, \({{\,\mathrm{reg}\,}}(S/I(G)) = 1\). We also provide a full description of the spectrum of \({{\,\mathrm{pd}\,}}(S/I(G))\) when \({{\,\mathrm{reg}\,}}(S/I(G))\) attains its minimum possible value 1.

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References

  1. 1.

    N. Boria, F. Della Croce and V. Th. Paschos, On the max min vertex cover problem, Disc. Appl. Math. 196 (2015), 62–71.

  2. 2.

    N. Bourgeois, F. Della Croce, B. Escoffier and V.Th. Paschos, Fast algorithms for min independent dominating set, Discrete Appl. Math. 161 (2013), 4-5, 558–572.

  3. 3.

    V. Costa, E. Haeusler, E.S. Laber and L. Nogueira, A note on the size of minimal covers, Info. Proc. Lett. 102 (2007), 124–126.

    MathSciNet  Article  Google Scholar 

  4. 4.

    H. Dao and J. Schweig, Projective dimension, graph domination parameters, and independence complex homology, J. of Combin. Theory, Ser. A 120 (2013), 453–469.

  5. 5.

    C.A. Francisco and A. Van Tuyl, Sequentially Cohen-Macaulay edge ideals, Proc. Amer. Math. Soc. 135 (2007), no. 8, 2327–2337.

    MathSciNet  Article  Google Scholar 

  6. 6.

    S. Gaspers, D. Kratsch and M. Liedloff, Exponential time algorithms for the minimum dominating set problem on some graph classes, in: L. Arge and R. Freivalds (Eds.), Proc. Scandinavian Workshop on Algorithm Theory, SWAT06, in: Lecture Notes in Computer Science, vol. 4059, Springer-Verlag, 2006, pp. 148–159.

  7. 7.

    S. Gaspers and M. Liedloff, A branch-and-reduce algorithm for finding a minimum independent dominating set in graphs, in: F.V. Fomin (Ed.), Proc. International Workshop on Graph Theoretical Concepts in Computer Science, WG06, in: Lecture Notes in Computer Science, vol. 4271, Springer-Verlag, 2006, pp. 78–89.

  8. 8.

    H. T. Hà and A. Van Tuyl, Monomial ideals, edge ideals of hypergraphs, and their graded Betti numbers, J. Algebraic Combin. 27 (2008), 215–245.

    MathSciNet  Article  Google Scholar 

  9. 9.

    M.M. Halldórsson, Approximating the minimum maximal independence number, Inform. Process. Lett. 46 (1993) 169–172.

    MathSciNet  Article  Google Scholar 

  10. 10.

    J. Herzog and T. Hibi, Monomial ideals, GTM 260, Springer, 2010.

  11. 11.

    M. Katzman, Characteristic-independence of Betti numbers of graph ideals, J. Combin. Theory Ser. A 113 (2006), 435–454.

    MathSciNet  Article  Google Scholar 

  12. 12.

    K. Kimura, Non-vanishingness of Betti numbers of edge ideals, in: Harmony of Gröbner Bases and the Modern Industrial Society (T. Hibi, Ed.), World Sci. Publ., 2012, pp. 153–168.

  13. 13.

    E. Miller and B. Sturmfels, Combinatorial Commutative Algebra, GTM 227, Springer, 2005.

    Google Scholar 

  14. 14.

    S. Morey and R.H. Villarreal, Edge ideals: algebraic and combinatorial properties, Progress in commutative algebra 1, 85–126, de Gruyter, Berlin, 2012.

    Google Scholar 

  15. 15.

    N. Terai, Alexander duality theorem and Stanley-Reisner rings, Free resolutions of coordinate rings of projective varieties and related topics (Japanese) (Kyoto, 1998), Sūrikaisekikenkyūsho Kōkyūroku 1078 (1999), 174–184.

    Google Scholar 

  16. 16.

    R.H. Villarreal, Cohen-Macaulay graphs, Manuscripta Math. 66 (1990), 277–293.

    MathSciNet  Article  Google Scholar 

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Acknowledgements

The first author was supported by Louisiana BOR grant LEQSF(2017-19)-ENH-TR-25, and the second author was supported by JSPS KAKENHI 19H00637. The authors thank Martin Milanic for informing us that the inequality (1.1) was already known and drawing our attention to [3], and thank Antonio Macchia for pointing out to us that \(C_4\) should be included in our classification result. The authors also thank anonymous referees for a careful read and useful suggestions.

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Correspondence to Huy Tài Hà.

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Hà, H.T., Hibi, T. MAX MIN Vertex Cover and the Size of Betti Tables. Ann. Comb. (2021). https://doi.org/10.1007/s00026-020-00521-4

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Keywords

  • Max min vertex cover
  • Min max independent set
  • Gap-free graph
  • Chordal graph
  • Regularity
  • Projective dimension
  • Betti table
  • Monomial ideal
  • Squarefree monomial
  • Edge ideal

Mathematics Subject Classification

  • 13D02
  • 05C70
  • 05E40