The Manickam-Mikl\'os-Singhi Property in Graphs and Hypergraphs

TL;DR

This paper investigates the Manickam-Miklós-Singhi property in graphs and hypergraphs, constructing new examples, analyzing probabilistic regimes, and extending conditions to higher uniformities.

Contribution

It introduces new families of regular graphs with the MMS property, extends conditions to hypergraphs of higher uniformity, and analyzes probabilistic regimes in random graphs.

Findings

Constructed new regular graphs with MMS property.

Identified regimes in Erdős–Rényi graphs where MMS holds with high probability.

Extended matching-based conditions to higher uniformity hypergraphs.

Abstract

This paper studies the Manickam-Mikl\'os-Singhi (MMS) property for graphs and hypergraphs. Using the structural characterisation of the $2$-uniform case, we construct new families of regular graphs with the MMS property. We then analyse the Erd\H{o}s--R\'enyi random graph model $\mathbf{G}(n,p)$ and identify regimes in which the MMS property holds with high probability. Finally, we extend the matching-based sufficient condition to higher uniformities via pseudo-matchings and introduce a blowout construction that produces higher-uniformity hypergraphs with the MMS property from lower-uniformity examples.