On the cover ideals of chordal graphs

The independence complex of a chordal graph is known to be shellable which is equivalent to the fact thatcover ideal of a chordal graph has linear quotients. We use this result to obtain recursive formulas for the Betti numbersof cover ideals of chordal graphs. Moreover, we give a new proof of such result which yields different shellings of theindependence complex.

PDF

___

[1] Dirac GA. On rigid circuit graphs. Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg 1961; 38: 71-76 (in German). doi: 10.1007/BF02992776
[2] Francisco C, Van Tuyl A. Sequentially Cohen-Macaulay edge ideals. Proceedings of the American Mathematical Society 2007; 135 (8): 2327-2337. doi: 10.1090/S0002-9939-07-08841-7
[3] Fröberg R. On Stanley-Reisner rings. Topics in Algebra, Part 2 Warsaw, Poland: Banach Center Publishing, 1990.
[4] Hà HT, Van Tuyl A. Monomial ideals, edge ideals of hypergraphs, and their graded Betti numbers. Journal of Algebraic Combinatorics 2008; 27: 215-245. doi: 10.1007/s10801-007-0079-y
[5] Herzog J, Hibi T. Monomial ideals. Graduate Texts in Mathematics, 260. London, UK: Springer-Verlag, 2011.
[6] Herzog J, Hibi T, Zheng X. Cohen-Macaulay chordal graphs. Journal of Combinatorial Theory Series A 2006; 113 (5): 911-916. doi: 10.1016/j.jcta.2005.08.007
[7] Hibi T, Kimura K, Murai S. Betti numbers of chordal graphs and f -vectors of simplicial complexes. Journal of Algebra 2010; 323 (6): 1678-1689. doi: 10.1016/j.jalgebra.2009.12.029
[8] Kimura K. Non-vanishingness of Betti Numbers of Edge Ideals. Harmony of Gröbner Bases and the Modern Industrial Society. New Jersey, USA: World Science Publishing, 2012.
[9] Sharifan L, Varbaro M. Graded Betti numbers of ideals with linear quotients. Matematiche (Catania) 2008; 63 (2): 257-265.
[10] Terai N. Alexander duality theorem and Stanley-Reisner rings. Free resolution of coordinate rings of projective varieties and related topics (Japanese). Surikaisekikenkyusho Kokyuroku 1999; 1078: 174-184.
[11] Van Tuyl A, Villarreal RH. Shellable graphs and sequentially Cohen-Macaulay bipartite graphs. Journal of Combinatorial Theory Series A. 2008; 115 (5): 799-814. doi: 10.1016/j.jcta.2007.11.001
[12] Woodroofe R. Matchings, coverings, and Castelnuovo-Mumford regularity. Journal of Commutative Algebra 2014; 6 (2): 287-304. doi: 10.1216/JCA-2014-6-2-287
[13] Zheng X. Resolutions of facet ideals. Communications in Algebra 2004; 32: 2301-2324. doi: 10.1081/AGB-120037222

ISSN: 1300-0098
Yayın Aralığı: Yılda 6 Sayı
Yayıncı: TÜBİTAK

Arşiv