Algebraic and combinatorial properties of ideals and algebras of uniform clutters of TDI systems

Luis A. Dupont () and Rafael H. Villarreal ()
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Luis A. Dupont: Centro de Investigación y de Estudios Avanzados del IPN
Rafael H. Villarreal: Centro de Investigación y de Estudios Avanzados del IPN

Journal of Combinatorial Optimization, 2011, vol. 21, issue 3, No 1, 269-292

Abstract: Abstract Let $\mathcal{C}$ be a uniform clutter and let A be the incidence matrix of $\mathcal{C}$ . We denote the column vectors of A by v 1,…,v q . Under certain conditions we prove that $\mathcal{C}$ is vertex critical. If $\mathcal{C}$ satisfies the max-flow min-cut property, we prove that A diagonalizes over ℤ to an identity matrix and that v 1,…,v q form a Hilbert basis. We also prove that if $\mathcal{C}$ has a perfect matching such that $\mathcal{C}$ has the packing property and its vertex covering number is equal to 2, then A diagonalizes over ℤ to an identity matrix. If A is a balanced matrix we prove that any regular triangulation of the cone generated by v 1,…,v q is unimodular. Some examples are presented to show that our results only hold for uniform clutters. These results are closely related to certain algebraic properties, such as the normality or torsion-freeness, of blowup algebras of edge ideals and to finitely generated abelian groups. They are also related to the theory of Gröbner bases of toric ideals and to Ehrhart rings.

Keywords: Uniform clutter; Max-flow min-cut; Normality; Rees algebra; Ehrhart ring; Balanced matrix; Edge ideal; Hilbert bases; Smith normal form; Unimodular regular triangulation (search for similar items in EconPapers)
Date: 2011
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DOI: 10.1007/s10878-009-9244-7

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