Matching
Allan (Yuping Lu)
Definitions
A matching of graph G is a subgraph of G such that every edge shares no vertex with
any other edge. That is, each vertex in matching M has degree one.
The size of a matching is the number of edges in that matching.
x1—x6, x2—x5 is a matching of size two
Definitions
A maximal matching is a matching M of a graph G with the property that if any edge not
in M is added to M, it is no longer a matching.
A maximum matching is a matching that contains the largest possible number of edges.
Definitions
A matching of a graph G is perfect if it contains all of G’s vertices.
An alternating path is a path that alternates between matching and non-matching
edges.
An augmenting path is an alternating path that starts and ends on unmatched vertices.
History
1914: Dénes Kőnig proves that every
regular bipartite graph has a perfect matching.
History
1935: Hall's marriage theorem, or simply
Hall's theorem, proved by Philip Hall.
History
1947: Tutte’s 1-factor Theorem named after
William Thomas Tutte
History
Berge's theorem proved by Claude Berge
in 1957
History
Jack Edmonds published the first polynomial
matching algorithm for nonbipartite graphs
and characterizes associated polytope in 1965.
Details can be found in [8].
Edmonds, Matching and the Birth of Polyhedral Combinatorics
History
1966: Paul Erdős and Alfréd Rényi
discover threshold function for
a random graph to have a perfect
matching.
History
Michael D. Plummer: A chronology of events in matching theory. [9]
Hall’s theorem
Let G = ((A,B),E) be a bipartite graph. G has a matching saturating A if and only if |N(S)| ≥
|S| for all S ⊆ A.
Hall’s theorem
Theorem based on Hall’s theorem
Let G = ((A,B),E) be a bipartite graph and k ∈ N. G has a matching of size k if and only if
|N(S)| ≥ |S|−|A|+k for any S ⊆ A.
Corollary: If G = ((A,B),E) is a k-regular bipartite graph (k ≥ 1), then G has a perfect
matching.
Tutte’s theorem
A graph, G = (V, E), has a perfect matching if and only if for every subset U of V, the
subgraph induced by V − U has at most |U| connected components with an odd number
of vertices
It is a generalization of Hall's marriage theorem from bipartite to arbitrary graphs.
Tutte’s theorem
Petersen's Theorem
If every vertex of G has degree 3 and G has no cut-edge, then G has a perfect matching.
Berge’s theorem
A matching M in a graph G is maximum if and only if there is no augmenting path with
M.
Generic Matching Algorithm
Initialization: M ← 0
Iteration: If there exists an M-augmenting path P, replace M ← M ⊕ P, i.e., by switching
free edges to matched edges and matched edges to free edges.
Edmond’s algorithm
How can one find an M-augmenting path?
Difficult in general Edmonds’ matching algorithm (Edmonds 1965)
Easy for bipartite graphs
Edmond’s algorithm
A blossom is defined as a cycle of odd length (2k + 1 edges) with k matching edges.
Edmond’s algorithm
Blossoms Lemma
Let G’ and M’ be obtained by contracting a blossom B in G to a single vertex. The
matching M of G is maximum iff M’ is maximum in G’.
Edmond’s algorithm
Detecting Blossoms
Performing the alternating path search. Label vertices at even distance from the root as
“outer”; Label vertices at odd distance from the root as “inner”.
If two outer vertices are found adjacent, we have a blossom.
Edmond’s algorithm
Applications
Dating services want to pair up compatible couples.
Interns need to be matched to hospital residency programs.
Other assignment problems involving resource allocation arise frequently, including
balancing the traffic load among servers on the Internet.
Open Problems
Matching polynomials of vertex transitive graphs
Author(s): Mohar
Conjecture: For every integer r there exists a vertex transitive graph G whose matching
polynomial has a root of multiplicity at least r.
Keywords: matching polynomial; vertex-transitive
Open Problems
The Berge-Fulkerson conjecture
Author(s): Berge; Fulkerson
Conjecture: If G is a bridgeless cubic graph, then there exist 6 perfect matchings M1,...,
M6 of G with the property that every edge of G is contained in exactly two of M1,...,M6.
Keywords: cubic; perfect matching
Find more on [7].
References
[1] http://mathworld.wolfram.com/Matching.html
[2] https://en.wikipedia.org/wiki/Matching_(graph_theory)
[3] http://www.ucl.ac.uk/~ucahbtw/docs/d1lesson4/matching_examples.pdf
[4] http://www.ucl.ac.uk/~ucahbtw/docs/d1lesson4/matchings_ben_notes.pdf
[5] http://www-math.mit.edu/~djk/18.310/Lecture-Notes/MatchingProblem.pdf
[6] https://www.cs.princeton.edu/courses/archive/fall06/cos341/handouts/graph2.pdf
References
[7] http://www.openproblemgarden.org/search/matching
[8] http://www.math.uiuc.edu/documenta/vol-ismp/34_pulleyblank-william.pdf
[9] M.D. Plummer Matching theory—a sampler: from Dénis König to the present Discrete
Math., 100 (1992), pp. 177–219
[10] https://en.wikipedia.org/wiki/Berge%27s_lemma
[11] https://en.wikipedia.org/wiki/Blossom_algorithm
[12] http://air.ug/~jquinn/teaching-files/graphtheory/graphtheory_week5.pdf
References
[13] http://www-sop.inria.fr/members/Frederic.Havet/Cours/matching.pdf
[14] https://en.wikipedia.org/wiki/Tutte_theorem
[15] https://www.math.hmc.edu/~kindred/cuc-only/math104/lectures/lect08.pdf
[16] https://en.wikipedia.org/wiki/Petersen%27s_theorem
[17] http://www.slideshare.net/akhayyat/maximum-matching-in-general-graphs
Homework
1. Six reporters Asif (A), Becky (B), Chris (C), David (D), Emma (E) and Fred (F), are to
be assigned to six news stories Business (1), Crime (2), Financial (3), Foreign(4),
Local (5) and Sport (6). The table shows possible allocations of reporters to news
stories. For example, Chris can be assigned to any one of stories 1, 2 or 4.
Homework
a) Show these possible allocations on a bipartite graph.
b) Show a maximal matching.
c) Does the graph has a perfect matching? If not, explain why.
2. Show that a tree has at most one perfect matching.
3. Let G be a graph on at least 2k +2 vertices which has a perfect matching. Show that if
every set of k independent edges is included in a perfect matching then every set of k −1
independent edges is included in a perfect matching.