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arXiv · 2309.15424

On Positive Matching Decomposition Conjectures of Hypergraphs

Marie Amalore Nambi·Neeraj Kumar
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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.

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Marie Amalore Nambi, Neeraj Kumar. 2023-09-27. On Positive Matching Decomposition Conjectures of Hypergraphs. https://doi.org/10.1142/s0218196725500390

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