Abstract
We study monomial cut ideals associated to a graph G, which are a monomial analogue of toric cut ideals as introduced by Sturmfels and Sullivant. Primary decompositions, projective dimensions, and Castelnuovo–Mumford regularities are investigated if the graph can be decomposed as 0-clique sums and disjoint union of subgraphs. The total Betti numbers of a cycle are computed. Moreover, we classify all Freiman ideals among monomial cut ideals.
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References
Auslander, M.: Modules over unramified regular local rings. Illinois J. Math. 5, 631–647 (1961)
Böröczky, K.J., Santos, F., Serra, O.: On sumsets and convex hull. Discrete Comput. Geom. 52, 705–729 (2014)
CoCoATeam, CoCoA: a system for doing Computations in Commutative Algebra. Available at http://cocoa.dima.unige.it
Deza, M., Laurent, M.: Applications of cut polyhedra I. J. Comput. Appl. Math. 55(2), 191–216 (1994)
Deza, M., Laurent, M.: Applications of cut polyhedra II. J. Comput. Appl. Math. 55(2), 217–247 (1994)
Deza, M., Laurent, M.: Geometry of cuts and metrics. In: Algorithms and Combinatorics 15, Springer (2010)
Diestel, R.: Graph theory. Fifth edition. In: Graduate Texts in Mathematics 173, Springer (2018)
Engström, A.: Cut ideals of \(K_4\)-minor-free graphs are generated by quadrics. Michigan Math. J. 60(3), 705–714 (2011)
Freiman, G.A.: Foundations of a structural theory of set addition. In: Translations of Mathematical Monographs 37, American Mathematical Society (1973)
Herzog, J.: Komplexe, Auflösungen und Dualität in der lokalen Algebra. Habilitation, Universität Regensburg (1974)
Herzog, J., Hibi, T.: Monomial ideals. In: Graduate Texts in Mathematics 260, Springer (2011)
Herzog, J., Hibi, T., Zhu, G.: The relevance of Freiman’s theorem for combinatorial commutative algebra. Math. Z. 291, 999–1014 (2019)
Karp, R.M.: Reducibility among combinatorial problems. In Proceedings of a symposium on the Complexity of computer computations, 85–103, (1972)
Koley, M., Römer, T.: Seminormality, canonical modules, and regularity of cut polytopes. J. Pure Appl. Algebra 226, 22 (2022)
Nagel, U., Petrović, S.: Properties of cut ideals associated to ring graphs. J. Commut. Algebra 1, 547–565 (2009)
Ohsugi, H.: Normality of cut polytopes of graphs is a minor closed property. Discrete Math. 310(6–7), 1160–1166 (2010)
Ohsugi, H.: Gorenstein cut polytopes. Eur. J. Combin. 38, 122–129 (2014)
Olteanu, A.: Monomial cut ideals. Comm. Algebra 41, 955–970 (2013)
Römer, T., Saeedi Madani, S.: Retracts and algebraic properties of cut algebras. Eur. J. Combin. 69, 214–236 (2018)
Römer, T., Saeedi Madani, S.: Cycle algebras and polytopes of matroids. Preprint (2021), arXiv:2105.00185
Sakamoto, R.: Lexicographic and reverse lexicographic quadratic Gröbner bases of cut ideals. J. Symbolic Comput. 103, 201–212 (2021)
Sturmfels, B., Sullivant, S.: Toric geometry of cuts and splits. Michigan Math. J. 57, 689–709 (2008)
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Funding was provided by Universität Duisburg-Essen (Grant No. 3149).
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Communicated by Isidoro Gitler.
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Herzog, J., Rahimbeigi, M. & Römer, T. Classes of cut ideals and their Betti numbers. São Paulo J. Math. Sci. 17, 172–187 (2023). https://doi.org/10.1007/s40863-022-00325-9
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DOI: https://doi.org/10.1007/s40863-022-00325-9