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

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 $a_{{\alpha }_1,\dots ,{\alpha }_d}$ of $A$ equals 1 if and only if $\left\{{\alpha }_1,\dots ,{\alpha }_d\right\}$ 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.

References in corpus (1)

• Upper bounds on the permanent of multidimensional (0,1)-matrices

Cited by in corpus (2)

• An asymptotic lower bound on the number of bent functions
• Toward a Three-dimensional Counterpart of Cruse's Theorem