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Robust Graph Ideals

Abstract

Let I be a toric ideal. We say I is robust if its universal Gröbner basis is a minimal generating set. We show that any robust toric ideal arising from a graph G is also minimally generated by its Graver basis. We then completely characterize all graphs which give rise to robust ideals. Our characterization shows that robustness can be determined solely in terms of graph-theoretic conditions on the set of circuits of G.

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References

  1. Bogart T., Hemmecke R., Petrović S.: Universal Gröbner bases of colored partition identities. Exp. Math. 21(4), 395–401 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  2. Boocher A., Robeva E.: Robust toric ideals. J. Symbolic Comput. 68(1), 254–264 (2015)

    Article  MATH  MathSciNet  Google Scholar 

  3. Gross E., Petrović S.: Combinatorial degree bound for toric ideals of hypergraphs. Internat. J. Algebra Comput. 23(6), 1503–1520 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  4. Ohsugi H., Hibi T.: Toric ideals generated by quadratic binomials. J. Algebra 218(2), 509–527 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  5. Reyes E., Tatakis C., Thoma A.: Minimal generators of toric ideals of graphs. Adv. Appl. Math. 48(1), 64–78 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  6. Sturmfels, B.: Gröbner Bases and Convex Polytopes. University Lecture Series 8. American Mathematical Society, Providence, RI (1996)

  7. Tatakis C., Thoma A.: On the universal Gröbner bases of toric ideals of graphs. J. Combin. Theory Ser. A 118(5), 1540–1548 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  8. Villarreal R.H.: Rees algebras of edge ideals. Comm. Algebra 23(9), 3513–3524 (1995)

    Article  MATH  MathSciNet  Google Scholar 

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Correspondence to Adam Boocher.

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Boocher, A., Brown, B.C., Duff, T. et al. Robust Graph Ideals. Ann. Comb. 19, 641–660 (2015). https://doi.org/10.1007/s00026-015-0288-3

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  • DOI: https://doi.org/10.1007/s00026-015-0288-3

Mathematics Subject Classification

  • 13P10
  • 05C25

Keywords

  • toric ideals
  • universal Gröbner bases
  • graph ideals