On the König–Hall–Egerváry theorem for multidimensional matrices and multipartite hypergraphs

Abstract

One of possible interpretations of the well-known König–Hall–Egerváry theorem is a full characterization of all bipartite graphs extremal for fractional matchings of a given weight (or, equivalently, a characterization of (0,1)-matrices extremal for partial fractional diagonals of a given length). In this paper we initiate the study of d-partite d-uniform hypergraphs that are extremal for fractional perfect matchings (or, equivalently, d-dimensional (0,1)-matrices that are extremal for polydiagonals). For this purpose, we analyze similarities and differences between 2-dimensional and multidimensional cases and put forward a series of questions and conjectures on properties of multidimensional extremal matrices (extremal hypergraphs). We also prove these conjectures for several parameters and provide a number of supporting constructions and examples.