Rees algebras and polyhedral cones of ideals of vertex covers of perfect graphs
Journal of Algebraic Combinatorics: An International Journal
Journal of Algebraic Combinatorics: An International Journal
| Hi-index | 0.00 |
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,驴,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 驴 to an identity matrix and that v 1,驴,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 驴 to an identity matrix. If A is a balanced matrix we prove that any regular triangulation of the cone generated by v 1,驴,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.