On the numbers of 1-factors and 1-factorizations of hypergraphs

TL;DR

This paper investigates the enumeration of 1-factors and 1-factorizations in uniform hypergraphs using permanents of their adjacency matrices, providing estimates for these quantities.

Contribution

It introduces a method to estimate the number of 1-factors and 1-factorizations in hypergraphs via permanents, extending combinatorial enumeration techniques.

Findings

Derived bounds for the number of 1-factors in hypergraphs.

Estimated the count of 1-factorizations in complete uniform hypergraphs.

Connected hypergraph properties with matrix permanents.

Abstract

A 1-factor of a hypergraph $G=(X,W)$ is a set of hyperedges such that every vertex of $G$ is incident to exactly one hyperedge from the set. A 1-factorization is a partition of all hyperedges of $G$ into disjoint 1-factors. The adjacency matrix of a $d$-uniform hypergraph $G$ is the $d$-dimensional (0,1)-matrix of order $|X|$ such that an element $a_{\alpha_1, \ldots, \alpha_d}$ of $A$ equals 1 if and only if $\left\{\alpha_1, \ldots, \alpha_d\right\}$ is a hyperedge of $G$. Here we estimate the number of 1-factors of uniform hypergraphs and the number of 1-factorizations of complete uniform hypergraphs by means of permanents of their adjacency matrices.