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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Bogart T., Hemmecke R., Petrović S.: Universal Gröbner bases of colored partition identities. Exp. Math. 21(4), 395–401 (2012)
Boocher A., Robeva E.: Robust toric ideals. J. Symbolic Comput. 68(1), 254–264 (2015)
Gross E., Petrović S.: Combinatorial degree bound for toric ideals of hypergraphs. Internat. J. Algebra Comput. 23(6), 1503–1520 (2013)
Ohsugi H., Hibi T.: Toric ideals generated by quadratic binomials. J. Algebra 218(2), 509–527 (1999)
Reyes E., Tatakis C., Thoma A.: Minimal generators of toric ideals of graphs. Adv. Appl. Math. 48(1), 64–78 (2012)
Sturmfels, B.: Gröbner Bases and Convex Polytopes. University Lecture Series 8. American Mathematical Society, Providence, RI (1996)
Tatakis C., Thoma A.: On the universal Gröbner bases of toric ideals of graphs. J. Combin. Theory Ser. A 118(5), 1540–1548 (2011)
Villarreal R.H.: Rees algebras of edge ideals. Comm. Algebra 23(9), 3513–3524 (1995)
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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
Mathematics Subject Classification
- toric ideals
- universal Gröbner bases
- graph ideals