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Partial coloring, vertex decomposability and sequentially Cohen-Macaulay simplicial complexes. (English) Zbl 1328.05207

Summary: In attempting to understand how combinatorial modifications alter algebraic properties of monomial ideals, several authors have investigated the process of adding “whiskers” to graphs. In this paper, we study a similar construction for building a simplicial complex \(\Delta_\chi\) from a coloring \(\chi\) of a subset of the vertices of \(\Delta\) and give necessary and sufficient conditions for this construction to produce vertex decomposable simplicial complexes. We apply this work to strengthen and give new proofs about sequentially Cohen-Macaulay edge ideals of graphs.

MSC:

05E45 Combinatorial aspects of simplicial complexes
05A15 Exact enumeration problems, generating functions
13C14 Cohen-Macaulay modules
13F55 Commutative rings defined by monomial ideals; Stanley-Reisner face rings; simplicial complexes
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References:

[1] J. Biermann and A. Van Tuyl, Balanced vertex decomposable simplicial complexes and their \(h\)-vectors . Electr. J. Comb. 20 (2013) #P15. · Zbl 1298.05332
[2] A. Björner, P. Frankl and R. Stanley, The number of faces of balanced Cohen-Macaulay complexes and a generalized Macaulay theorem , Combinatorica 7 (1987), 23-34. · Zbl 0651.05010
[3] A. Björner and M. Wachs, Shellable nonpure complexes and posets I, Trans. Amer. Math. Soc. 348 (1996), 1299-1327. · Zbl 0857.05102
[4] CoCoATeam, CoCoA: A system for doing computations in commutative algebra , available at http://cocoa.dima.unige.it.
[5] D. Cook, II, Simplicial decomposability , JSAG 2 (2010), 20-23. · Zbl 1311.05002
[6] D. Cook, II and U. Nagel, Cohen-Macaulay graphs and face vectors of flag complexes , SIAM J. Discr. Math. 26 (2012), 89-101. · Zbl 1245.05138
[7] A. Dochtermann and A. Engström, Algebraic properties of edge ideals via combinatorial topology , Electr. J. Combin. 16 (2009), Research Paper 2, 24 pages. · Zbl 1161.13013
[8] C. Francisco and H.T. Hà, Whiskers and sequentially Cohen-Macaulay graphs , J. Comb. Theor. 115 (2008), 304-316. · Zbl 1142.13021
[9] A. Frohmader, How to construct a flag complex with a given face vector , preprint (2011 arXiv:
[10] D.R. Grayson and M.E. Stillman, Macaulay2, A software system for research in algebraic geometry , http://www.math.uiuc.edu/Macaulay2/.
[11] J. Herzog and T. Hibi, Monomial ideals , Grad. Texts Math. 260 , Springer-Verlag, New York, 2011. · Zbl 1206.13001
[12] J. Jonsson, Simplicial complexes of graphs , Lect. Notes Math. 1928 , Springer-Verlag, Berlin, 2008. · Zbl 1152.05001
[13] J. Provan and L. Billera, Decompositions of simplicial complexes related to the diameters of convex polyhedra , Math. Oper. Res. 5 (1980), 576–594. · Zbl 0457.52005
[14] A. Van Tuyl, Sequentially Cohen-Macaulay bipartite graphs : Vertex decomposability and regularity , Arch. Math. 93 (2009), 451-459. · Zbl 1184.13062
[15] R.H. Villarreal, Cohen-Macaulay graphs , Manuscr. Math. 66 (1990), 277–293. · Zbl 0737.13003
[16] R. Woodroofe, Vertex decomposable graphs and obstructions to shellability , Proc. Amer. Math. Soc. 137 (2009), 3235-3246. · Zbl 1180.13031
[17] —-, Chordal and sequentially Cohen-Macaulay clutters , Electr. J. Comb. 18 (2011), Paper 208, 20 pages. · Zbl 1236.05213
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