Comptes Rendus
no. 12
Combinatorics/Topology
Connectivity of pseudomanifold graphs from an algebraic point of view

Presented by: the Editorial Board

Karim A. Adiprasito1, 2; Afshin Goodarzi3; Matteo Varbaro4
1 Institut des hautes études scientifiques, Bures-sur-Yvette, France
2 Einstein Institute for Mathematics, Hebrew University of Jerusalem, Jerusalem, Israel
3 Department of Mathematics, Kungliga Tekniska Högskolan, S-100 44 Stockholm, Sweden
4 Dipartimento di Matematica, Università di Genova, Via Dodecaneso 35-16146, Genova, Italy
Comptes Rendus. Mathématique, Volume 353 (2015) no. 12, pp. 1061-1065.

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.

La connexité des graphes des complexes simpliciaux et polytopaux est un sujet classique remontant au moins à Steinitz. Il a été étudié depuis par de nombreux auteurs, dont Balinski, Barnette, Athanasiadis et Björner. Dans cette note, nous présentons une approche unifiée nous permettant d'obtenir des résultats plus généraux. De plus, nous faisons un lien avec l'algèbre commutative en rapprochant les problèmes de connexité des nombres de Betti gradués des anneaux de Stanley–Reisner associés.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2015.09.018

Karim A. Adiprasito 1, 2; Afshin Goodarzi 3; Matteo Varbaro 4

1 Institut des hautes études scientifiques, Bures-sur-Yvette, France
2 Einstein Institute for Mathematics, Hebrew University of Jerusalem, Jerusalem, Israel
3 Department of Mathematics, Kungliga Tekniska Högskolan, S-100 44 Stockholm, Sweden
4 Dipartimento di Matematica, Università di Genova, Via Dodecaneso 35-16146, Genova, Italy 
@article{CRMATH_2015__353_12_1061_0,
     author = {Karim A. Adiprasito and Afshin Goodarzi and Matteo Varbaro},
     title = {Connectivity of pseudomanifold graphs from an algebraic point of view},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {1061--1065},
     publisher = {Elsevier},
     volume = {353},
     number = {12},
     year = {2015},
     doi = {10.1016/j.crma.2015.09.018},
     language = {en},
}
TY  - JOUR
AU  - Karim A. Adiprasito
AU  - Afshin Goodarzi
AU  - Matteo Varbaro
TI  - Connectivity of pseudomanifold graphs from an algebraic point of view
JO  - Comptes Rendus. Mathématique
PY  - 2015
SP  - 1061
EP  - 1065
VL  - 353
IS  - 12
PB  - Elsevier
DO  - 10.1016/j.crma.2015.09.018
LA  - en
ID  - CRMATH_2015__353_12_1061_0
ER  - 
%0 Journal Article
%A Karim A. Adiprasito
%A Afshin Goodarzi
%A Matteo Varbaro
%T Connectivity of pseudomanifold graphs from an algebraic point of view
%J Comptes Rendus. Mathématique
%D 2015
%P 1061-1065
%V 353
%N 12
%I Elsevier
%R 10.1016/j.crma.2015.09.018
%G en
%F CRMATH_2015__353_12_1061_0
Karim A. Adiprasito; Afshin Goodarzi; Matteo Varbaro. Connectivity of pseudomanifold graphs from an algebraic point of view. Comptes Rendus. Mathématique, Volume 353 (2015) no. 12, pp. 1061-1065. doi : 10.1016/j.crma.2015.09.018. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2015.09.018/

[1] C.A. Athanasiadis Some combinatorial properties of flag simplicial pseudomanifolds and spheres, Ark. Mat., Volume 49 (2011) no. 1, pp. 17-29

[2] D. Barnette Decompositions of homology manifolds and their graphs, Isr. J. Math., Volume 41 (1982) no. 3, pp. 203-212

[3] A. Björner; M. Las Vergnas; B. Sturmfels; N. White; G.M. Ziegler Oriented Matroids, Encyclopedia of Mathematics and Its Applications, vol. 46, Cambridge University Press, Cambridge, UK, 1999

[4] A. Björner; K. Vorwerk On the connectivity of manifold graphs, Proc. Amer. Math. Soc., Volume 143 (2015) no. 10, pp. 4123-4132

[5] A. Goodarzi Clique vectors of k-connected chordal graphs, J. Comb. Theory, Ser. A, Volume 132 (2015), pp. 188-193

[6] J. Herzog; T. Hibi Monomial Ideals, Graduate Texts in Mathematics, vol. 260, Springer-Verlag London Ltd., London, 2011

[7] E. Miller; B. Sturmfels Combinatorial Commutative Algebra, Graduate Texts in Mathematics, vol. 227, Springer-Verlag, New York, 2005

[8] E. Steinitz Polyeder und Raumeinteilungen (W.F. Meyer; H. Mohrmann, eds.), Encyklopädie der mathematischen Wissenschaften, Dritter Band: Geometrie, III.1.2., Heft 9, Kapitel III A B 12, B. G. Teubner, Leipzig, Germany, 1922, pp. 1-139

[9] G.M. Ziegler Lectures on Polytopes, Graduate Texts in Mathematics, vol. 152, Springer, New York, 1995 (revised edition, 1998, seventh updated printing 2007)

Cited by Sources:

Comments - Policy