Geodesic
On the diameter of dual graphs of Stanley-Reisner rings and Hirsch type bounds on abstractions of polytopes
Brent Holmes1
1University of Kansas
The electronic journal of combinatorics, Tome 25 (2018) no. 1
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Let $R$ be an equidimensional commutative Noetherian ring of positive dimension. The dual graph $\mathcal{G} (R)$ of $R$ is defined as follows: the vertices are the minimal prime ideals of $R$, and the edges are the pairs of prime ideals $(P_1,P_2)$ with height$(P_1 + P_2) = 1$. If $R$ satisfies Serre's property $(S_2)$, then $\mathcal{G} (R)$ is connected. In this note, we provide lower and upper bounds for the maximum diameter of dual graphs of Stanley-Reisner rings satisfying $(S_2)$. These bounds depend on the number of variables and the dimension. Dual graphs of $(S_2)$ Stanley-Reisner rings are a natural abstraction of the $1$-skeletons of polyhedra. We discuss how our bounds imply new Hirsch-type bounds on $1$-skeletons of polyhedra.
DOI : 10.37236/6831
Classification : 13D02, 13F55, 05C12, 05E45
Mots-clés : simplicial complex, dual graph, Stanley-Reisner ring, Serre condition, Hirsch conjecture, polyhedra

Brent Holmes  1

1 University of Kansas 
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     title = {On the diameter of dual graphs of {Stanley-Reisner} rings and {Hirsch} type bounds on abstractions of polytopes},
     journal = {The electronic journal of combinatorics},
     year = {2018},
     volume = {25},
     number = {1},
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     url = {http://geodesic.mathdoc.fr/articles/10.37236/6831/}
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Brent Holmes. On the diameter of dual graphs of Stanley-Reisner rings and Hirsch type bounds on abstractions of polytopes. The electronic journal of combinatorics, Tome 25 (2018) no. 1. doi: 10.37236/6831

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