Hall's marriage theorem HallsMarriageTheorem 2002-04-16 01:14:21 2003-09-18 13:40:01 Theorem % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here Let $S = \{ S_1,S_2,\dots S_n \}$ be a finite collection of finite sets. There exists a system of distinct representatives of $S$ if and only if the following condition holds for any $T \subseteq S$: $$\left | \cup T \right | \geq |T|$$ As a corollary, if this condition fails to hold anywhere, then no SDR exists. This is known as Hall's marriage theorem. The name arises from a particular application of this theorem. Suppose we have a finite set of single men/women, and, for each man/woman, a finite collection of women/men to whom this person is attracted. An SDR for this collection would be a way each man/woman could be (theoretically) married happily. Hence, Hall's marriage theorem can be used to determine if this is possible. An application of this theorem to graph theory gives that if $G(V_1, V_2, E)$ is a bipartite graph, then $G$ has a complete matching that saturates every vertex of $V_1$ if and only if $|S|\leq |N(S)|$ for every subset $S\subset V_1$.