On positive matching decomposition conjectures of hypergraphs

In this paper, we prove the conjectures of Gharakhloo and Welker [S. Gharakhloo and V. Welker, Hypergraph LSS-ideals and coordinate sections of symmetric tensors, Commun. Algebra 51(8) (2023) 3299–3309, Conjectures 3.5 and 3.6] that the positive matching decomposition number [Formula: see text] of a [Formula: see text]-uniform hypergraph is bounded from above by a polynomial of degree [Formula: see text] in terms of the number of vertices. Moreover, we derive a lower bound for [Formula: see text] specifically for complete [Formula: see text]-uniform hypergraphs. Additionally, we obtain an upper bound for [Formula: see text] of r-uniform hypergraphs. As an application from an algebraic point of view, we obtain the radical, complete intersection, and prime properties of Lovász–Saks–Schrijver (LSS) ideals of r-uniform hypergraphs. For an r-uniform hypergraphs [Formula: see text] such that [Formula: see text] for all [Formula: see text], 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 properties of LSS ideals.