Splittings of toric ideals of graphs

2024/05/16 by Anargyros Katsabekis, Katsabekis, Anargyros, Apostolos Thoma +1 · 2 citations
Mathematics · #05C25 #05E40 #13F65 #14M25 #Commutative Algebra (math.AC) #Commutative Algebra and Its Applications #FOS: Mathematics

paper · pdf · doi:10.48550/arxiv.2405.09836

openalex publication_date 2024/05/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let G be a simple graph on the vertex set \v₁,…,v_n\. An algebraic object attached to G is the toric ideal I_G. We say that I_G is subgraph splittable if there exist subgraphs G₁ and G₂ of G such that I_G=I_G1+I_G2, where both I_G1 and I_G2 are not equal to I_G. We show that I_G is subgraph splittable if and only if it is edge splittable. We also prove that the toric ideal of a complete bipartite graph is not subgraph splittable. In contrast, we show that the toric ideal of a complete graph K_n is always subgraph splittable when n ≥ 4. Additionally, we show that the toric ideal of K_n has a minimal splitting if and only if 4 ≤ n ≤ 5. Finally, we prove that any minimal splitting of I_G is also a reduced splitting.

Cited by

• Splittings of Ideals of Points in ℙ¹×ℙ¹
• Toric intersections

Related

• Toric splittings
• Toric geometry of cuts and splits
• Toric ideals and diagonal 2-minors
• Splitting cycles in graphs
• Minimal generators of toric ideals of graphs