A balanced non-partitionable Cohen–Macaulay complex
Algebraic Combinatorics, Volume 2 (2019) no. 6, p. 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 : 2018-01-29
Revised : 2019-03-02
Accepted : 2019-04-06
Published online : 2019-12-04
DOI : https://doi.org/10.5802/alco.78
Classification:  05E45,  13F55
Keywords: simplicial complex, balancedness, Cohen–Macaulay, partitionability
@article{ALCO_2019__2_6_1149_0,
     author = {Juhnke-Kubitzke, Martina and Venturello, Lorenzo},
     title = {A balanced non-partitionable Cohen--Macaulay complex},
     journal = {Algebraic Combinatorics},
     publisher = {MathOA foundation},
     volume = {2},
     number = {6},
     year = {2019},
     pages = {1149-1157},
     doi = {10.5802/alco.78},
     language = {en},
     url = {https://alco.centre-mersenne.org/item/ALCO_2019__2_6_1149_0}
}
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/item/ALCO_2019__2_6_1149_0/

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

[2] Bruns, Winfried; Herzog, Jürgen Cohen–Macaulay rings, Cambridge Studies in Advanced Mathematics, Volume 39, Cambridge University Press, Cambridge, 1993, xii+403 pages | MR 1251956 | Zbl 0788.13005

[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 | Article | MR 3519473 | Zbl 1341.05256

[4] Garsia, Adriano M. Combinatorial methods in the theory of Cohen–Macaulay rings, Adv. in Math., Volume 38 (1980) no. 3, pp. 229-266 | Article | MR 597728 | Zbl 0461.06002

[5] Hachimori, Masahiro Constructible complexes and recursive division of posets, Theoret. Comput. Sci., Volume 235 (2000) no. 2, pp. 225-237 | Article | MR 1756122 | Zbl 0938.68882

[6] Hachimori, Masahiro Decompositions of two-dimensional simplicial complexes, Discrete Math., Volume 308 (2008) no. 11, pp. 2307-2312 | Article | MR 2404561 | Zbl 1137.52006

[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 | Article | MR 0304376 | Zbl 0233.14010

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

[9] Reisner, Gerald Allen Cohen–Macaulay quotients of polynomial rings, Advances in Math., Volume 21 (1976) no. 1, pp. 30-49 | Article | MR 0407036 | Zbl 0345.13017

[10] Rudin, Mary Ellen An unshellable triangulation of a tetrahedron, Bull. Amer. Math. Soc., Volume 64 (1958), p. 90-91 | Article | MR 0097055 | Zbl 0082.37602

[11] Stanley, Richard P. The upper bound conjecture and Cohen–Macaulay rings, Studies in Appl. Math., Volume 54 (1975) no. 2, pp. 135-142 | Article | MR 0458437 | Zbl 0308.52009

[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 0572989 | Zbl 0376.55007

[13] Stanley, Richard P. Balanced Cohen–Macaulay complexes, Trans. Amer. Math. Soc., Volume 249 (1979) no. 1, pp. 139-157 | Article | MR 526314 | Zbl 0411.05012

[14] Stanley, Richard P. Combinatorics and commutative algebra, Progress in Mathematics, Volume 41, Birkhäuser Boston, Inc., Boston, MA, 1996, x+164 pages | MR 1453579 | Zbl 0838.13008

[15] Ziegler, Günter M. Shelling polyhedral 3-balls and 4-polytopes, Discrete Comput. Geom., Volume 19 (1998) no. 2, pp. 159-174 | Article | MR 1600042 | Zbl 0898.52006