Toric ideals associated with gap-free graphs
Abstract
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 Gröbner 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.
- Publication:
-
arXiv e-prints
- Pub Date:
- June 2014
- DOI:
- 10.48550/arXiv.1406.6634
- arXiv:
- arXiv:1406.6634
- Bibcode:
- 2014arXiv1406.6634D
- Keywords:
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- Mathematics - Commutative Algebra;
- Mathematics - Combinatorics;
- 13P10;
- 05E40
- E-Print:
- 13 pages, v2. To appear in Journal of Pure and Applied Algebra