On Positive Matching Decomposition Conjectures of Hypergraphs

TL;DR

This paper proves conjectures about the positive matching decomposition number of 3-uniform hypergraphs, establishing polynomial bounds and characterizations, advancing understanding of hypergraph matchings and algebraic properties.

Contribution

It confirms conjectures on pmd bounds for 3-uniform hypergraphs, provides bounds for r-uniform hypergraphs, and characterizes positive matchings via strong alternate closed walks.

Findings

pmd of 3-uniform hypergraphs is polynomially bounded

lower bounds for complete 3-uniform hypergraphs

characterization of positive matchings in certain hypergraphs

Abstract

In this paper, we prove the conjectures of Gharakhloo and Welker (2023) that the positive matching decomposition number (pmd) of a $3$-uniform hypergraph is bounded from above by a polynomial of degree $2$ in terms of the number of vertices. Moreover, we derive a lower bound for pmd specifically for complete $3$-uniform hypergraphs. Additionally, we obtain an upper bound for pmd of $r$-uniform hypergraphs. For a $r$-uniform hypergraphs $H=(V,E)$ such that $\lvert e_i\cap e_j\rvert \leq 1$ for all $e_i,e_j \in E$, we give a characterization of positive matching in terms of strong alternate closed walks. For a specific class of hypergraphs, we classify the radical and complete intersection Lov\'{a}sz$-$Saks$-$Schrijver ideals.