Depth of initial ideals of normal edge rings

Let \(G\) be a finite graph on the vertex set \([d] = \{1, ..., d \}\) with the edges \(e_1, ..., e_n\) and \(K[\tb] = K[t_1, ..., t_d]\) the polynomial ring in \(d\) variables over a field \(K\). The edge ring of \(G\) is the semigroup ring \(K[G]\) which is generated by those monomials \(\tb^e = t...

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Bibliographic Details
Published in:arXiv.org 2011-01
Main Authors: Hibi, Takayuki, Higashitani, Akihiro, Kimura, Kyouko, O'Keefe, Augustine B
Format: Article
Language:English
Subjects:
Graph theory
Homomorphisms
Integers
Polynomials
Rings (mathematics)
Online Access:Get full text
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Summary:Let \(G\) be a finite graph on the vertex set \([d] = \{1, ..., d \}\) with the edges \(e_1, ..., e_n\) and \(K[\tb] = K[t_1, ..., t_d]\) the polynomial ring in \(d\) variables over a field \(K\). The edge ring of \(G\) is the semigroup ring \(K[G]\) which is generated by those monomials \(\tb^e = t_it_j\) such that \(e = \{i, j\}\) is an edge of \(G\). Let \(K[\xb] = K[x_1, ..., x_n]\) be the polynomial ring in \(n\) variables over \(K\) and define the surjective homomorphism \(\pi : K[\xb] \to K[G]\) by setting \(\pi(x_i) = \tb^{e_i}\) for \(i = 1, ..., n\). The toric ideal \(I_G\) of \(G\) is the kernel of \(\pi\). It will be proved that, given integers \(f\) and \(d\) with \(6 \leq f \leq d\), there exist a finite connected nonbipartite graph \(G\) on \([d]\) together with a reverse lexicographic order \(
ISSN:2331-8422