Regularity and projective dimension of the edge ideal of 𝐶₅-free vertex decomposable graphs

Khosh-Ahang, Fahimeh; Moradi, Somayeh

doi:10.1090/S0002-9939-2014-11906-X

Regularity and projective dimension of the edge ideal of $C_5$-free vertex decomposable graphs
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Proc. Amer. Math. Soc. 142 (2014), 1567-1576 Request permission

Abstract:

In this paper, we explain the regularity, projective dimension and depth of the edge ideal of some classes of graphs in terms of invariants of graphs. We show that for a $C_5$-free vertex decomposable graph $G$, $\mbox {reg}(R/I(G))= c_G$, where $c_G$ is the maximum number of $3$-disjoint edges in $G$. Moreover, for this class of graphs we characterize $\mbox {pd}(R/I(G))$ and $\mbox {depth}(R/I(G))$. As a corollary we describe these invariants in forests and sequentially Cohen-Macaulay bipartite graphs.

References

Anders Björner and Michelle L. Wachs, Shellable nonpure complexes and posets. I, Trans. Amer. Math. Soc. 348 (1996), no. 4, 1299–1327. MR 1333388, DOI 10.1090/S0002-9947-96-01534-6
Anders Björner and Michelle L. Wachs, Shellable nonpure complexes and posets. II, Trans. Amer. Math. Soc. 349 (1997), no. 10, 3945–3975. MR 1401765, DOI 10.1090/S0002-9947-97-01838-2
Anton Dochtermann and Alexander Engström, Algebraic properties of edge ideals via combinatorial topology, Electron. J. Combin. 16 (2009), no. 2, Special volume in honor of Anders Björner, Research Paper 2, 24. MR 2515765
Hailong Dao and Jay Schweig, Projective dimension, graph domination parameters, and independence complex homology, J. Combin. Theory Ser. A 120 (2013), no. 2, 453–469. MR 2995051, DOI 10.1016/j.jcta.2012.09.005
David Eisenbud, Commutative algebra, Graduate Texts in Mathematics, vol. 150, Springer-Verlag, New York, 1995. With a view toward algebraic geometry. MR 1322960, DOI 10.1007/978-1-4612-5350-1
Christopher A. Francisco and Adam Van Tuyl, Sequentially Cohen-Macaulay edge ideals, Proc. Amer. Math. Soc. 135 (2007), no. 8, 2327–2337. MR 2302553, DOI 10.1090/S0002-9939-07-08841-7
Ralf Fröberg, On Stanley-Reisner rings, Topics in algebra, Part 2 (Warsaw, 1988) Banach Center Publ., vol. 26, PWN, Warsaw, 1990, pp. 57–70. MR 1171260
Huy Tài Hà and Adam Van Tuyl, Monomial ideals, edge ideals of hypergraphs, and their graded Betti numbers, J. Algebraic Combin. 27 (2008), no. 2, 215–245. MR 2375493, DOI 10.1007/s10801-007-0079-y
Jürgen Herzog, Takayuki Hibi, and Xinxian Zheng, Dirac’s theorem on chordal graphs and Alexander duality, European J. Combin. 25 (2004), no. 7, 949–960. MR 2083448, DOI 10.1016/j.ejc.2003.12.008
Kyouko Kimura, Non-vanishingness of Betti numbers of edge ideals, Harmony of Gröbner bases and the modern industrial society, World Sci. Publ., Hackensack, NJ, 2012, pp. 153–168. MR 2986877, DOI 10.1142/9789814383462_{0}009
Fatemeh Mohammadi and Somayeh Moradi, Resolution of unmixed bipartite graphs, preprint. arXiv:0901.3015V1 (2009).
Susan Morey and Rafael H. Villarreal, Edge ideals: algebraic and combinatorial properties, Progress in commutative algebra 1, de Gruyter, Berlin, 2012, pp. 85–126. MR 2932582
J. Scott Provan and Louis J. Billera, Decompositions of simplicial complexes related to diameters of convex polyhedra, Math. Oper. Res. 5 (1980), no. 4, 576–594. MR 593648, DOI 10.1287/moor.5.4.576
Naoki Terai, Alexander duality theorem and Stanley-Reisner rings, Sūrikaisekikenkyūsho K\B{o}kyūroku 1078 (1999), 174–184. Free resolutions of coordinate rings of projective varieties and related topics (Japanese) (Kyoto, 1998). MR 1715588
Adam Van Tuyl, Sequentially Cohen-Macaulay bipartite graphs: vertex decomposability and regularity, Arch. Math. (Basel) 93 (2009), no. 5, 451–459. MR 2563591, DOI 10.1007/s00013-009-0049-9
Adam Van Tuyl and Rafael H. Villarreal, Shellable graphs and sequentially Cohen-Macaulay bipartite graphs, J. Combin. Theory Ser. A 115 (2008), no. 5, 799–814. MR 2417022, DOI 10.1016/j.jcta.2007.11.001
Rafael H. Villarreal, Monomial algebras, Monographs and Textbooks in Pure and Applied Mathematics, vol. 238, Marcel Dekker, Inc., New York, 2001. MR 1800904
Rafael H. Villarreal, Cohen-Macaulay graphs, Manuscripta Math. 66 (1990), no. 3, 277–293. MR 1031197, DOI 10.1007/BF02568497
Russ Woodroofe, Vertex decomposable graphs and obstructions to shellability, Proc. Amer. Math. Soc. 137 (2009), no. 10, 3235–3246. MR 2515394, DOI 10.1090/S0002-9939-09-09981-X
Xinxian Zheng, Resolutions of facet ideals, Comm. Algebra 32 (2004), no. 6, 2301–2324. MR 2100472, DOI 10.1081/AGB-120037222