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.

Revised:

Accepted:

Published online:

Keywords: geometric vertex decomposition, toric ideals of graphs, liaison

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

@article{ALCO_2023__6_4_965_0, author = {Cummings, Mike and Da Silva, Sergio and Rajchgot, Jenna and Van Tuyl, Adam}, title = {Geometric vertex decomposition and liaison for toric ideals of graphs}, journal = {Algebraic Combinatorics}, pages = {965--997}, publisher = {The Combinatorics Consortium}, volume = {6}, number = {4}, year = {2023}, doi = {10.5802/alco.295}, language = {en}, url = {https://alco.centre-mersenne.org/articles/10.5802/alco.295/} }

TY - JOUR AU - Cummings, Mike AU - Da Silva, Sergio AU - Rajchgot, Jenna AU - Van Tuyl, Adam TI - Geometric vertex decomposition and liaison for toric ideals of graphs JO - Algebraic Combinatorics PY - 2023 SP - 965 EP - 997 VL - 6 IS - 4 PB - The Combinatorics Consortium UR - https://alco.centre-mersenne.org/articles/10.5802/alco.295/ DO - 10.5802/alco.295 LA - en ID - ALCO_2023__6_4_965_0 ER -

%0 Journal Article %A Cummings, Mike %A Da Silva, Sergio %A Rajchgot, Jenna %A Van Tuyl, Adam %T Geometric vertex decomposition and liaison for toric ideals of graphs %J Algebraic Combinatorics %D 2023 %P 965-997 %V 6 %N 4 %I The Combinatorics Consortium %U https://alco.centre-mersenne.org/articles/10.5802/alco.295/ %R 10.5802/alco.295 %G en %F ALCO_2023__6_4_965_0

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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