# Matching

A matching, also called an independent edge phối, on a graph

is a set of edges of
such that no two sets share a vertex in common.

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It is not possible for a matching on a graph with

nodes lớn exceed
edges. When a matching with
edges exists, it is called a perfect matching. When a matching exists that leaves a single vertex unmatched, it is called a near-perfect matching.

While not all graphs have perfect matchings, a largest matching (commonly known as a maximum matching or maximum independent edge set) exists for every graph. The kích thước of this maximum matching is called the matching number of

and is denoted
.

The number of matchings in a graph is sometimes called the Hosoyaindex.

A maximal independent edge mix, which is different from a maximum independent edge set, is a matching that cannot be enlarged by simply adding an edge. Such matchings are easy khổng lồ compute, but are not necessarily maximum independent edge sets. A maximal independent edge set on a general graph can be found using MaximalMatching in the cheap-kenya-vacation-tips.com Language package Combinatorica` , but not using a using built-in function in the cheap-kenya-vacation-tips.com Language.

The blossom algorithm can be used lớn find a maximum independent edge mix in a general graph, while the simpler Hungarian maximum matching algorithm can be used for bipartite graphs. A maximum independent edge mix can be computed in the cheap-kenya-vacation-tips.com Language using FindIndependentEdgeSet.

Let the number of distinct

-matchings of a graph with
vertices be denoted
. Then
(since the empty set consisting of no edges is always a 0-matching) and
, where
is the edge count of
.

The matching polynomial is defined by

và the matching-generating polynomialby

The numbers of distinct

-matchings for various specials classes of graphs are summarized in the following table, where
denotes a factorial,
is a double factorial,
is a binomial coefficient, and
is the discrete delta function.

 graph complete graph complete bipartite graph cycle graph empty graph path graph

SEE ALSO: Berge"s Theorem, Blossom Algorithm, Hosoya Index, Hungarian Maximum Matching Algorithm, Independent Edge Set, Marriage Theorem, Matching-Generating Polynomial, Matching Number, Matching Polynomial, Maximal Independent Edge Set, Maximum Independent Edge Set, Near-Perfect Matching, Perfect Matching, Stable Marriage Problem
REFERENCES:

Hopcroft, J. & Karp, R. "An

Algorithm for Maximum Matching in Bipartite Graphs." SIAM J. Comput. 2, 225-231, 1975.

Lovász, L. và Plummer, M.D. MatchingTheory. Amsterdam, Netherlands: North-Holl&, 1986.

Pemmaraju, S. and Skiemãng cầu, S. "Matching." §8.4 in Computational Discrete Mathematics: Combinatorics and Graph Theory in Mathematica. Cambridge, England: Cambridge University Press, pp.343-351, 2003.

Skiemãng cầu, S. "Matching." §6.4 in Implementing Discrete Mathematics: Combinatorics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, pp.240-246, 1990.

Zwiông xã, U. "Lecture Notes on: Maximum Matching in Bipartite and Non-BipartiteGraphs." 2009. https://www.cs.tau.ac.il/~zwick/grad-algo-0910/match.pdf.

Referenced on cheap-kenya-vacation-tips.com|Alpha: Matching
CITE THIS AS:

Weisstein, Eric W. "Matching." From cheap-kenya-vacation-tips.com--A cheap-kenya-vacation-tips.com Web Resource. https://cheap-kenya-vacation-tips.com/Matching.html

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