Toric ideals associated with gap-free graphs
Journal of Pure and Applied Algebra 219 (2015), pp. 3862-3872 In this article we prove that every toric ideal associated with a gap-free graph $G$ has a squarefree lexicographic initial ideal. Moreover, in the particular case when the complementary graph of $G$ is chordal (i.e. when the edge ideal o...
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| Format: | Journal Article |
| Language: | English |
| Published: |
11-12-2014
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| Online Access: | Get full text |
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| Summary: | Journal of Pure and Applied Algebra 219 (2015), pp. 3862-3872 In this article we prove that every toric ideal associated with a gap-free
graph $G$ has a squarefree lexicographic initial ideal. Moreover, in the
particular case when the complementary graph of $G$ is chordal (i.e. when the
edge ideal of $G$ has a linear resolution), we show that there exists a reduced
Gr\"obner basis $\mathcal{G}$ of the toric ideal of $G$ such that all the
monomials in the support of $\mathcal{G}$ are squarefree. Finally, we show
(using work by Herzog and Hibi) that if $I$ is a monomial ideal generated in
degree 2, then $I$ has a linear resolution if and only if all powers of $I$
have linear quotients, thus extending a result by Herzog, Hibi and Zheng. |
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| DOI: | 10.48550/arxiv.1406.6634 |