ALGEBRAIC COMBINATORICS

no. 6
A balanced non-partitionable Cohen–Macaulay complex
Juhnke-Kubitzke, Martina1; Venturello, Lorenzo1
1 Universität Osnabrück Institut für Mathematik Albrechtstraße 28a 49076 Osnabrück, Germany
Algebraic Combinatorics, Volume 2 (2019) no. 6, pp. 1149-1157.

In a recent article, Duval, Goeckner, Klivans and Martin disproved the longstanding conjecture by Stanley, that every Cohen–Macaulay simplicial complex is partitionable. We construct counterexamples to this conjecture that are even balanced, i.e. their underlying graph has a minimal coloring. This answers a question by Duval et al. in the negative.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/alco.78
Classification: 05E45, 13F55
Keywords: simplicial complex, balancedness, Cohen–Macaulay, partitionability
Juhnke-Kubitzke, Martina 1; Venturello, Lorenzo 1

1 Universität Osnabrück Institut für Mathematik Albrechtstraße 28a 49076 Osnabrück, Germany
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
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Juhnke-Kubitzke, Martina; Venturello, Lorenzo. A balanced non-partitionable Cohen–Macaulay complex. Algebraic Combinatorics, Volume 2 (2019) no. 6, pp. 1149-1157. doi : 10.5802/alco.78. https://alco.centre-mersenne.org/articles/10.5802/alco.78/

[1] Ball, Michael O. Network reliability analysis: algorithms and complexity, Ph. D. Thesis, Cornell University (USA) (1977)

[2] Bruns, Winfried; Herzog, Jürgen Cohen–Macaulay rings, Cambridge Studies in Advanced Mathematics, 39, Cambridge University Press, Cambridge, 1993, xii+403 pages | MR | Zbl

[3] Duval, Art M.; Goeckner, Bennet; Klivans, Caroline J.; Martin, Jeremy L. A non-partitionable Cohen–Macaulay simplicial complex, Adv. Math., Volume 299 (2016), pp. 381-395 | DOI | MR | Zbl

[4] Garsia, Adriano M. Combinatorial methods in the theory of Cohen–Macaulay rings, Adv. in Math., Volume 38 (1980) no. 3, pp. 229-266 | DOI | MR | Zbl

[5] Hachimori, Masahiro Constructible complexes and recursive division of posets, Theoret. Comput. Sci., Volume 235 (2000) no. 2, pp. 225-237 | DOI | MR | Zbl

[6] Hachimori, Masahiro Decompositions of two-dimensional simplicial complexes, Discrete Math., Volume 308 (2008) no. 11, pp. 2307-2312 | DOI | MR | Zbl

[7] Hochster, Melvin Rings of invariants of tori, Cohen–Macaulay rings generated by monomials, and polytopes, Ann. of Math. (2), Volume 96 (1972) no. 2, pp. 318-337 | DOI | MR | Zbl

[8] Provan, John Scott Decompositions, shelling and diameters of simplicial complexes and convex polyhedra, Ph. D. Thesis, Cornell University (USA) (1977), 128 pages

[9] Reisner, Gerald Allen Cohen–Macaulay quotients of polynomial rings, Advances in Math., Volume 21 (1976) no. 1, pp. 30-49 | DOI | MR | Zbl

[10] Rudin, Mary Ellen An unshellable triangulation of a tetrahedron, Bull. Amer. Math. Soc., Volume 64 (1958), pp. 90-91 | DOI | MR | Zbl

[11] Stanley, Richard P. The upper bound conjecture and Cohen–Macaulay rings, Studies in Appl. Math., Volume 54 (1975) no. 2, pp. 135-142 | DOI | MR | Zbl

[12] Stanley, Richard P. Cohen–Macaulay complexes, Higher combinatorics (Proc. NATO Advanced Study Inst., Berlin, 1976) (NATO Adv. Study Inst. Ser., Ser. C: Math. and Phys. Sci.), Volume 31, Reidel, Dordrecht, 1977, pp. 51-62 | MR | Zbl

[13] Stanley, Richard P. Balanced Cohen–Macaulay complexes, Trans. Amer. Math. Soc., Volume 249 (1979) no. 1, pp. 139-157 | DOI | MR | Zbl

[14] Stanley, Richard P. Combinatorics and commutative algebra, Progress in Mathematics, 41, Birkhäuser Boston, Inc., Boston, MA, 1996, x+164 pages | MR | Zbl

[15] Ziegler, Günter M. Shelling polyhedral 3-balls and 4-polytopes, Discrete Comput. Geom., Volume 19 (1998) no. 2, pp. 159-174 | DOI | MR | Zbl

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