Classes of cut ideals and their Betti numbers

TL;DR

This paper investigates the algebraic properties of monomial cut ideals related to graphs, focusing on their decompositions, Betti numbers, and classification of Freiman ideals, extending understanding of their algebraic and combinatorial structure.

Contribution

It introduces a detailed study of monomial cut ideals, including their decompositions, Betti numbers, and classification of Freiman ideals, advancing the algebraic understanding of graph-related ideals.

Findings

Betti numbers of cycles are explicitly computed.

Primary decompositions and regularities are characterized for graphs decomposed as $0$-clique sums.

All Freiman ideals among monomial cut ideals are classified.

Abstract

We study monomial cut ideals associated to a graph $G$, which are a monomial analogue of toric cut ideals as introduced by Sturmfels and Sullivant. Primary decompositions, projective dimensions, and Castelnuovo-Mumford regularities are investigated if the graph can be decomposed as $0$-clique sums and disjoint union of subgraphs. The total Betti numbers of a cycle are computed. Moreover, we classify all Freiman ideals among monomial cut ideals.