ALGEBRAIC COMBINATORICS

no. 4
Geometric vertex decomposition and liaison for toric ideals of graphs
Cummings, Mike1; Da Silva, Sergio2; Rajchgot, Jenna1; Van Tuyl, Adam1
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.

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:
Revised:
Accepted:
Published online:
DOI: 10.5802/alco.295
Classification: 13P10, 14M25, 05E40, 13C04
Keywords: geometric vertex decomposition, toric ideals of graphs, liaison
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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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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