Geometric vertex decomposition and liaison for toric ideals of graphs

Cummings, Mike; Da Silva, Sergio; Rajchgot, Jenna; Van Tuyl, Adam

doi:10.5802/alco.295

Cummings, Mike ¹; Da Silva, Sergio ²; Rajchgot, Jenna ¹; Van Tuyl, Adam ¹

¹ McMaster University Department of Mathematics and Statistics 1280 Main St W Hamilton ON CAN L8S4L8
² Virginia State University Department of Mathematics and Economics 1 Hayden Dr Petersburg VA USA 23806

Algebraic Combinatorics, Volume 6 (2023) no. 4, pp. 965-997.

Abstract

Geometric vertex decomposability for polynomial ideals is an ideal-theoretic generalization of vertex decomposability for simplicial complexes. Indeed, a homogeneous geometrically vertex decomposable ideal is radical and Cohen-Macaulay, and is in the Gorenstein liaison class of a complete intersection (glicci).

In this paper, we initiate an investigation into when the toric ideal $I_{G}$ of a finite simple graph $G$ is geometrically vertex decomposable. We first show how geometric vertex decomposability behaves under tensor products, which allows us to restrict to connected graphs. We then describe a graph operation that preserves geometric vertex decomposability, thus allowing us to build many graphs whose corresponding toric ideals are geometrically vertex decomposable. Using work of Constantinescu and Gorla, we prove that toric ideals of bipartite graphs are geometrically vertex decomposable. We also propose a conjecture that all toric ideals of graphs with a square-free degeneration with respect to a lexicographic order are geometrically vertex decomposable. As evidence, we prove the conjecture in the case that the universal Gröbner basis of $I_{G}$ is a set of quadratic binomials. We also prove that some other families of graphs have the property that $I_{G}$ is glicci.

Received: 2022-08-11
Revised: 2023-01-18
Accepted: 2023-01-21
Published online: 2023-08-29

DOI: 10.5802/alco.295

Classification: 13P10, 14M25, 05E40, 13C04
Keywords: geometric vertex decomposition, toric ideals of graphs, liaison

Author's affiliations:

Cummings, Mike ¹; Da Silva, Sergio ²; Rajchgot, Jenna ¹; Van Tuyl, Adam ¹

¹ McMaster University Department of Mathematics and Statistics 1280 Main St W Hamilton ON CAN L8S4L8
² Virginia State University Department of Mathematics and Economics 1 Hayden Dr Petersburg VA USA 23806

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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     title = {Geometric vertex decomposition and liaison for toric ideals of graphs},
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Cummings, Mike; Da Silva, Sergio; Rajchgot, Jenna; Van Tuyl, Adam. Geometric vertex decomposition and liaison for toric ideals of graphs. Algebraic Combinatorics, Volume 6 (2023) no. 4, pp. 965-997. doi : 10.5802/alco.295. https://alco.centre-mersenne.org/articles/10.5802/alco.295/

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