Abstract
Let G be a simple graph on the vertex set \({\{1,\ldots,n\}}\) with m edges. An algebraic object attached to G is the ideal P G generated by diagonal 2-minors of an \({n \times n}\) matrix of variables. We prove that if G is bipartite, then every initial ideal of P G is generated by squarefree monomials of degree at most \({\lfloor{\frac{m+n+1}{2}} \rfloor}\). Furthermore, we completely characterize all connected graphs G for which P G is the toric ideal associated to a finite simple graph. Finally we compute in certain cases the universal Gröbner basis of P G .
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This work was carried out during the tenure of an ERCIM “Alain Bensoussan” Fellowship Programme. The research leading to these results has received funding from the European Union Seventh Framework Programme (FP7/2007-2013) under grant agreement no. 246016.
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Katsabekis, A. Toric ideals and diagonal 2-minors. Acta Math. Hungar. 150, 83–98 (2016). https://doi.org/10.1007/s10474-016-0651-9
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DOI: https://doi.org/10.1007/s10474-016-0651-9