A bound on degrees of primitive elements of toric graph ideals

Rychlewicz, Kamil

doi:10.48550/arxiv.1701.07137

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2018

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“…Among them, the set of primitive binomials, which is known to form the Graver basis, was studied widely related to a problem initiated by Sturmfels, called true degree problem. (See [22,24,25,27] for detail.) For a toric ideal I A , an irreducible binomial B " B `´B ´of I A is primitive if there exists no other binomial B 0 " B 0 ´B0 such that B 0 |B `and B 0 |B ´.…”

Section: Introductionmentioning

confidence: 99%

On toric ideals arising from signed graphs

Kim¹,

Park²

2018

Preprint

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A signed graph is a pair pG, τ q of a graph G and its sign τ , where a sign τ is a function from tpe, vq | e P EpGq, v P V pGq, v P eu to t1, ´1u. Note that graphs or digraphs are special cases of signed graphs. In this paper, we study the toric ideal I pG,τ q associated with a signed graph pG, τ q, and the results of the paper give a unified idea to explain some known results on the toric ideals of a graph or a digraph. We characterize all primitive binomials of I pG,τ q , and then focus on the complete intersection property. More precisely, we find a complete list of graphs G such that I pG,τ q is a complete intersection for every sign τ .

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Section: Introductionmentioning

confidence: 99%