Connectivity of pseudomanifold graphs from an algebraic point of view. (Connexité des graphes de pseudo-variétés d’un point de vue algébrique.) (English. French summary) Zbl 1331.05119
Summary: The connectivity of graphs of simplicial and polytopal complexes is a classical subject going back at least to Steinitz, and the topic has since been studied by many authors, including Balinski, Barnette, Athanasiadis, and Björner. In this note, we provide a unifying approach that allows us to obtain more general results. Moreover, we provide a relation to commutative algebra by relating connectivity problems to graded Betti numbers of the associated Stanley-Reisner rings.
MSC:
| 05C40 | Connectivity |
| 05C25 | Graphs and abstract algebra (groups, rings, fields, etc.) |
| 05E45 | Combinatorial aspects of simplicial complexes |
References:
| [1] | Athanasiadis, C. A., Some combinatorial properties of flag simplicial pseudomanifolds and spheres, Ark. Mat., 49, 1, 17-29 (2011) · Zbl 1235.52020 |
| [2] | Barnette, D., Decompositions of homology manifolds and their graphs, Isr. J. Math., 41, 3, 203-212 (1982) · Zbl 0498.57004 |
| [3] | Björner, A.; Las Vergnas, M.; Sturmfels, B.; White, N.; Ziegler, G. M., Oriented Matroids, Encyclopedia of Mathematics and Its Applications, vol. 46 (1999), Cambridge University Press: Cambridge University Press Cambridge, UK · Zbl 0944.52006 |
| [4] | Björner, A.; Vorwerk, K., On the connectivity of manifold graphs, Proc. Amer. Math. Soc., 143, 10, 4123-4132 (2015) · Zbl 1320.05141 |
| [5] | Goodarzi, A., Clique vectors of \(k\)-connected chordal graphs, J. Comb. Theory, Ser. A, 132, 188-193 (2015) · Zbl 1307.05177 |
| [6] | Herzog, J.; Hibi, T., Monomial Ideals, Graduate Texts in Mathematics, vol. 260 (2011), Springer-Verlag London Ltd.: Springer-Verlag London Ltd. London · Zbl 1206.13001 |
| [7] | Miller, E.; Sturmfels, B., Combinatorial Commutative Algebra, Graduate Texts in Mathematics, vol. 227 (2005), Springer-Verlag: Springer-Verlag New York · Zbl 1090.13001 |
| [8] | Steinitz, E., Polyeder und Raumeinteilungen, (Meyer, W. F.; Mohrmann, H., Encyklopädie der mathematischen Wissenschaften, Dritter Band: Geometrie, III.1.2., Heft 9, Kapitel III A B 12 (1922), B. G. Teubner: B. G. Teubner Leipzig, Germany), 1-139 |
| [9] | Ziegler, G. M., Lectures on Polytopes, Graduate Texts in Mathematics, vol. 152 (1995), Springer: Springer New York, revised edition, 1998, seventh updated printing 2007 · Zbl 0823.52002 |
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