Bounds on the regularity of toric ideals of graphs

Biermann, Jennifer; O'Keefe, Augustine B.; Tuyl, Adam Van

doi:10.1016/j.aam.2016.11.003

Cited by 30 publications

(39 citation statements)

References 32 publications

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“…. , v n ] given by ϕ(e i ) = v i 1 v i 2 where e i = {v i 1 , v i 2 } ∈ E. Some properties of the homological invariants of I G can be found in [2,3,4,6,7,11,18,21]. Our main result adds to this list of properties, and contributes to Hibi and Matsuda's program.…”

Section: Introductionmentioning

confidence: 71%

Regularity and h-polynomials of toric ideals of graphs

Favacchio¹,

Keiper²,

Tuyl³

2020

Preprint

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For all integers 4 ≤ r ≤ d, we show that there exists a finite simple graph G = G r,d with toric ideal I G ⊂ R such that R I G has (Castelnuovo-Mumford) regularity r and h-polynomial of degree d. To achieve this goal, we identify a family of graphs such that the graded Betti numbers of the associated toric ideal agree with its initial ideal, and furthermore, this initial ideal has linear quotients. As a corollary, we can recover a result of Hibi, Higashitani, Kimura, and O'Keefe that compares the depth and dimension of toric ideals of graphs.

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Section: Introductionmentioning

confidence: 71%

Regularity and h-polynomials of toric ideals of graphs

Favacchio¹,

Keiper²,

Tuyl³

2020

Preprint

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“…x γ : γ ∈ Γ} remains a linearly independent set of monomials. This implies that the remainder of x 1 g by the division by the Gröbner basis of the ideal (x 2 i − x 2 j : i, j ∈ V G ) is not zero, contradicting (1). Hence, we must have g = 0.…”

Section: Introductionmentioning

confidence: 92%

“…Consider the graph, G, in Figure 2. It is a Hamiltonian bipartite graph, with Hamiltonian cycle (1,2,5,4,3,6,1). This cycle can be taken as the ear P 1 of an ear decomposition starting from P 0 = 1.…”

Section: Regularitymentioning

confidence: 99%

“…In the case of graphs, these ideals include, but are not limited to, the toric ideal, the edge ideal and binomial edge ideal, and, in this framework, one of the algebraic invariants that has been the object of growing interest is the Castelnuovo-Mumford regularity. Bounds for the regularity of the toric ideal have been obtained in [1,10]. These bounds involve the number and sizes of families of disjoint induced complete bipartite subgraphs of the graph.…”

Section: Introductionmentioning

confidence: 99%

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Joins, Ears and Castelnuovo-Mumford regularity

Neves¹,

Pinto²,

Villarreal³

2019

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We introduce a new class of polynomial ideals associated to a simple graph, G. Let K[E G ] be the polynomial ring on the edges of G and K[V G ] the polynomial ring on the vertices of G. We associate to G an ideal, I(X G ), defined as the preimage of (xwhich sends a variable, te, associated to an edge e = {i, j}, to the product x i x j of the variables associated to its vertices. We show that K[E G ]/I(X G ) is a one-dimensional, Cohen-Macaulay, graded ring, that I(X G ) is a binomial ideal and that, with respect to a fixed monomial order, its initial ideal has a generating set independent of the field K. We focus on the Castelnuovo-Mumford regularity of I(X G ) providing the following sharp upper and lower bounds:where µ(G) is the maximum vertex join number of the graph and b 0 (G) is the number of its connected components. We show that the lower bound is attained for a bipartite graph and use this to derive a new combinatorial result on the number of even length ears of nested ear decomposition.

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“…The approach of using the initial ideal was also employed in [2]; we hope this case study further illustrates the usefulness of this technique to compute (or bound) graded Betti numbers of toric ideals, especially since we know of very few families of graphs where we know all the values β i,j (I G ). Our work is in the spirit of Conca and Varbaro's paper [6] which shows that many of the homological invariants of an ideal are related to its initial ideal if the initial ideal is square-free.…”

mentioning

confidence: 99%