Editing Hall's marriage theorem (section)

==Applications==
The theorem has many applications. For example, for a [[Playing card|standard deck of cards]], dealt into 13 piles of 4 cards each, the marriage theorem implies that it is possible to select one card from each pile so that the selected cards contain exactly one card of each rank (Ace, 2, 3, ..., Queen, King). This can be done by constructing a bipartite graph with one partition containing the 13 piles and the other partition containing the 13 ranks. The remaining proof follows from the marriage condition. More generally, any [[Regular graph|regular]] bipartite graph has a perfect matching.<ref>{{cite web| last = DeVos |first = Matt | url = http://www.sfu.ca/~mdevos/notes/graph/345_matchings.pdf | title = Graph Theory |website = Simon Fraser University}}</ref>{{rp|2}}

More abstractly, let <math>G</math> be a [[Group (mathematics)|group]], and <math>H</math> be a finite [[Index of a subgroup|index]] [[subgroup]] of <math>G</math>. Then the marriage theorem can be used to show that there is a set <math>T</math> such that <math>T</math> is a transversal for both the set of left [[coset]]s and right cosets of <math>H</math> in <math>G</math>.<ref>{{cite journal|last1 = Button|first1 = Jack |first2= Maurice |last2 = Chiodo| first3 = Mariano |last3 = Zeron-Medina Laris|title = Coset Intersection Graphs for Groups|journal =  The American Mathematical Monthly|volume = 121|issue =  10|year = 2014|pages =  922–26|doi = 10.4169/amer.math.monthly.121.10.922|arxiv = 1304.6111 |s2cid = 16417209 |quote = For <math>H</math> a finite index subgroup of <math>G</math>, the existence of a left-right transversal is well known, sometimes presented as an application of Hall’s marriage theorem.}}</ref>

The marriage theorem is used in the usual proofs of the fact that an <math>r\times n</math> [[Latin rectangle]] can always be extended to an <math>(r+1)\times n</math> Latin rectangle when <math>r<n</math>, and so, ultimately to a [[Latin square]].<ref>{{cite journal | title = An existence theorem for latin squares|first = Marshall|last =  Hall|
journal = Bull. Amer. Math. Soc. |volume = 51 |year = 1945 |issue = 6|pages = 387–388|doi = 10.1090/S0002-9904-1945-08361-X|doi-access = free}}</ref>