Cohen-Macaulay clutters with combinatorial optimization properties and parallelizations of normal edge ideals

TL;DR

This paper explores the properties of uniform Cohen-Macaulay clutters, demonstrating how certain combinatorial optimization properties are preserved under specific transformations and their implications for edge ideals and related conjectures.

Contribution

It establishes that the packing and max-flow min-cut properties can be extended to Cohen-Macaulay minors and proves the normality of edge ideals is preserved under parallelizations.

Findings

Cohen-Macaulay minors inherit packing and max-flow min-cut properties.

Normality of edge ideals is preserved under parallelizations.

Applications to conjectures and optimization problems in combinatorics.

Abstract

Let C be a uniform clutter and let I=I(C) be its edge ideal. We prove that if C satisfies the packing property (resp. max-flow min-cut property), then there is a uniform Cohen-Macaulay clutter C1 satisfying the packing property (resp. max-flow min-cut property) such that C is a minor of C1. For arbitrary edge ideals of clutters we prove that the normality property is closed under parallelizations. Then we show some applications to edge ideals and clutters which are related to a conjecture of Conforti and Cornu\'ejols and to max-flow min-cut problems.