A generalized marriage theorem

TL;DR

This paper generalizes Hall's marriage theorem by introducing the concept of disparate selections in set-valued mappings on graphs, providing necessary and sufficient conditions, and applying the results to Latin squares and classical theorems.

Contribution

It introduces the concept of disparate kernels and selections, extending the classical marriage theorem to more general set-valued mappings on graphs.

Findings

Established necessary and sufficient conditions for disparate selections.

Defined and computed the disparate kernel of set-valued mappings.

Connected the generalized theorem to Latin squares and Hall's theorem.

Abstract

We consider a set-valued mapping on a simple graph and ask for the existence of a disparate selection. The term disparate is defined in the paper and we present a sufficient and necessary condition for the existence of a disparate selection. This approach generalizes the classical marriage theorem of Hall. We define the disparate kernel of the set-valued mapping and provide calculation methods for the disparate kernel and a disparate selection. Our main theorem is applied to a result of Ryser on the completion of partially prepopulated Latin squares and we derive Hall's marriage theorem.