Rees algebras and polyhedral cones of ideals of vertex covers of perfect graphs
Rafael H. Villarreal
Centro de Investigación y de Estudios Avanzados del IPN Departamento de Matemáticas Apartado Postal 14-740 07000 Mexico City DF Mexico
DOI: 10.1007/s10801-007-0088-x
Abstract
Let G be a perfect graph and let J be its ideal of vertex covers. We show that the Rees algebra of J is normal and that this algebra is Gorenstein if G is unmixed. Then we give a description-in terms of cliques-of the symbolic Rees algebra and the Simis cone of the edge ideal of G.
Pages: 293–305
Keywords: keywords perfect graphs; normality; edge ideals; symbolic Rees algebras; standard Gorenstein algebras; max-flow min-cut; clutters; simis cone; Hilbert basis; totally dual integral
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References
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2. Berge, C.: Balanced matrices. Math. Program. 2(1), 19-31 (1972)
3. Bruns, W., Herzog, J.: Cohen-Macaulay Rings. Cambridge University Press, Cambridge (1997), Re- vised edition
4. Bruns, W., Koch, R.: NORMALIZ, computing normalizations of affine semigroups (2003). Available from: ftp.mathematik.Uni-Osnabrueck.DE/pub/osm/kommalg/software
5. Chvátal, V.: On certain polytopes associated with graphs. J. Comb. Theory Ser. B 18, 138-154 (1975)
6. Cornuéjols, G.: Combinatorial Optimization: Packing and Covering. CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 74, SIAM (2001)
7. Diestel, R.: Graph Theory, 2nd edn. Graduate Texts in Mathematics, vol.
173. Springer, New York (2000)
8. Escobar, C., Martínez-Bernal, J., Villarreal, R.H.: Relative volumes and minors in monomial subrings. Linear Algebra Appl. 374, 275-290 (2003)
9. Escobar, C., Villarreal, R.H., Yoshino, Y.: Torsion freeness and normality of blowup rings of monomial ideals. In: Commutative Algebra. Lecture Notes Pure Appl. Math., vol. 244, pp. 69-84. Chapman & Hall/CRC, Boca Raton (2006)
10. Gitler, I., Reyes, E., Villarreal, R.H.: Blowup algebras of square-free monomial ideals and some links to combinatorial optimization problems. Rocky Mt. J. Math. 37(4) (2007, to appear)
11. Gitler, I., Valencia, C., Villarreal, R.H.: A note on Rees algebras and the MFMC property. Beiträge Algebra Geom. 48(1), 141-150 (2007)
12. Herzog, J., Hibi, T., Trung, N.V.: Symbolic powers of monomial ideals and vertex cover algebras.