In graph theory, perfect matching in high-degree hypergraphs is a research avenue trying to find sufficient conditions for existence of a perfect matching in a hypergraph, based only on the degree of vertices or subsets of them.
Introduction
Degrees and matchings in graphs In a simple graph G = (V, E), the degree of a vertex v, often denoted by deg(v) or δ(v), is the number of edges in E adjacent to v. The minimum degree of a graph, often denoted by deg(G) or δ(v), is the minimum of deg(v) over all vertices v in V. A matching in a graph is a set of edges such that each vertex is adjacent to at most one edge; a perfect matching is a matching in which each vertex is adjacent to exactly one edge. A perfect matching does not always exist, and thus it is interesting to find sufficient conditions that guarantee its existence. One such condition follows from Dirac's theorem on Hamiltonian cycles. It says that, if deg(G) ≥ n⁄2, then the graph admits a Hamiltonian cycle; this implies that it admits a perfect matching. The factor n⁄2 is tight, since the complete bipartite graph on (n⁄2 – 1, n⁄2 + 1) vertices has degree n⁄2 – 1 but does not admit a perfect matching. The results described below aim to extend these results from graphs to hypergraphs.
Degrees in hypergraphs In a hypergraph H = (V, E), each edge of E may contain more than two vertices of V. The degree of a vertex v in V is, as before, the number of edges in E that contain v. But in a hypergraph we can also consider the degree of subsets of vertices: given a subset U of V, deg(U) is the number of edges in E that contain all vertices of U. Thus, the degree of a hypergraph can be defined in different ways depending on the size of subsets whose degree is considered. Formally, for every integer d ≥ 1, degd(H) is the minimum of deg(U) over all subsets U of V that contain exactly d vertices. Thus, deg1(H) corresponds to the definition of a degree of a simple graph, namely the smallest degree of a single vertex; deg2(H) is the smallest degree of a pair of vertices; etc. A hypergraph H = (V, E) is called r-uniform if every hyperedge in E contains exactly r vertices of V. In r-uniform graphs, the relevant values of d are 1, 2, … , r – 1. In a simple graph, r = 2.
Conditions on 1-vertex degree Several authors proved sufficient conditions for the case d = 1, i.e., conditions on the smallest degree of a single vertex.
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