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Matching which covers every node of the graph
In graph theory, a perfect matching in a graph is a matching that covers every vertex of the graph. More formally, given a graph G with edges E and vertices
Perfect_matching
Set of edges without common vertices
matching is a maximum matching. The following figure shows examples of maximum matchings in the same three graphs. A perfect matching is a matching that
Matching_(graph_theory)
Characterization of graphs with perfect matchings
with perfect matchings. It is a special case of the Tutte–Berge formula. The goal is to characterize all graphs that do not have a perfect matching. Start
Tutte's theorem on perfect matchings
Tutte's_theorem_on_perfect_matchings
Set of hyperedges where every pair is disjoint
the natural extension of the notion of perfect matching in a graph. A fractional matching M is called perfect if for every vertex v in V, the sum of fractions
Matching_in_hypergraphs
Result in combinatorics and graph theory
theoretic formulation answers whether a finite bipartite graph has a perfect matching—that is, a way to match each vertex from one group uniquely to an adjacent
Hall's_marriage_theorem
Refinement of perfect matching theorems
in graph theory that is used to refine various theorems related to perfect matching in graphs, such as Hall's marriage theorem. This was first studied
Deficiency_(graph_theory)
Mathematical graph theorem
follows: Petersen's Theorem. Every cubic, bridgeless graph contains a perfect matching. In other words, if a graph has exactly three edges at each vertex
Petersen's_theorem
Tool used in probabilistic polynomial identity testing
contain a perfect matching? Theorem 2 (Tutte 1947): A Tutte matrix determinant is not a 0-polynomial if and only if there exists a perfect matching. A subset
Schwartz–Zippel_lemma
new perfect matching that contains e. Hence, e is maximally matchable. Conversely, if e is maximally matchable, then it is in some perfect matching N.
Maximally_matchable_edge
Approximation for the travelling salesman problem
handshaking lemma, O has an even number of vertices. Find a minimum-weight perfect matching M in the subgraph induced in G by O. Combine the edges of M and T to
Christofides_algorithm
Area of research in mathematics (graph theory)
theory, perfect matching in high-degree hypergraphs is a research avenue trying to find sufficient conditions for existence of a perfect matching in a hypergraph
Perfect matching in high-degree hypergraphs
Perfect_matching_in_high-degree_hypergraphs
Generalizations in graph theory
guaranteeing that a bipartite graph (X + Y, E) admits a perfect matching, or - more generally - a matching that saturates all vertices of Y. The condition involves
Hall-type theorems for hypergraphs
Hall-type_theorems_for_hypergraphs
Polynomial-time algorithm for the assignment problem
cost of each perfect matching is at least the value of each potential. This can be seen by first noticing that the total cost of the matching is the sum
Hungarian_algorithm
Shape representing matchings in a graph
In graph theory, the matching polytope of a given graph is a geometric object representing the possible matchings in the graph. It is a convex polytope
Matching_polytope
On bipartite matching and vertex cover
graph has a perfect matching, and more generally that the chromatic index of any bipartite graph (that is, the minimum number of matchings into which it
Kőnig's theorem (graph theory)
Kőnig's_theorem_(graph_theory)
Graph of n vertices with a perfect matching for every subgraph of n-1 vertices
perfect matching, a way of grouping the remaining vertices into adjacent pairs. A matching of all but one vertex of a graph is called a near-perfect matching
Factor-critical_graph
Combinatorial optimization problem
assignment, and the graph-theoretic version is called minimum-cost perfect matching. Otherwise, it is called unbalanced assignment. If the total cost of
Assignment_problem
largest fractional matching can be larger than the largest integral matching. For example, a 3-cycle admits a perfect fractional matching of size 3 2 {\displaystyle
Fractional_matching
Partition of a graph into spanning subgraphs
k-factorable if it admits a k-factorization. In particular, a 1-factor is a perfect matching, and a 1-factorization of a k-regular graph is a proper edge coloring
Graph_factorization
Graph divided into two independent sets
formed from complete bipartite graphs by removing the edges of a perfect matching. Hypercube graphs, partial cubes, and median graphs are bipartite.
Bipartite_graph
Cubic graph with 10 vertices and 15 edges
as minors. The K5 minor can be formed by contracting the edges of a perfect matching, for instance the five short edges in the first picture. The K3,3 minor
Petersen_graph
Problem of grouping into triples
graph theory, a 3-dimensional matching is a generalization of bipartite matching (also known as 2-dimensional matching) to 3-partite hypergraphs, which
3-dimensional_matching
Robustness of graph perfect matchings
of all perfect matchings or near-perfect matchings (matchings that cover all but one vertex in a graph with an odd number of vertices). Matching preclusion
Matching_preclusion
are a subclass of the perfect graphs. 3. A perfect matching is a matching that saturates every vertex; see matching. 4. A perfect 1-factorization is a
Glossary_of_graph_theory
example, in graph theory to obtain upper bounds for the number of perfect matchings in a bipartite graph. The permanent of a square binary matrix A =
Bregman–Minc_inequality
Tool for working with matrices
a perfect matching. Birkhoff's algorithm is a greedy algorithm: it greedily finds perfect matchings and removes them from the fractional matching. It
Birkhoff_algorithm
Mathematical function
vertices a, b, c, and d has three perfect matchings: ab and cd, ac and bd, and ad and bc. Perfect matchings may be described in several other equivalent
Double_factorial
Polynomial of the elements of a matrix
{\displaystyle x_{i}} to vertex y j {\displaystyle y_{j}} . If the weight of a perfect matching σ {\displaystyle \sigma } that matches x i {\displaystyle x_{i}} to
Permanent_(mathematics)
Subset of a graph's edges
a matching. In particular, it is a perfect matching: a matching M in which every vertex is incident with exactly one edge in M. A perfect matching (if
Edge_cover
Geometric construct
union of two unit squares meeting edge-to-edge. Equivalently, it is a perfect matching in the grid graph formed by placing a vertex at the center of each
Domino_tiling
be used to count the perfect matchings of the graph. This is the main idea behind the FKT algorithm for counting perfect matchings in planar graphs, which
Pfaffian_orientation
Edge-colored graph matching where all edges have distinct colors
are only two perfect matchings, and each of them is colored by a single color. This invokes the question: when does a large rainbow matching is guaranteed
Rainbow_matching
Topics referred to by the same term
4-vertex-connected planar graphs Tutte's theorem on perfect matchings, a characterization of the graphs having perfect matchings Tutte's spring theorem, on the planarity
Tutte's_theorem
Algorithm for counting perfect matchings in planar graphs
the number of perfect matchings in a planar graph in polynomial time. This same task is #P-complete for general graphs. For matchings that are not required
FKT_algorithm
Assignment of colors to edges of a graph
the multigraph case. A matching in a graph G is a set of edges, no two of which are adjacent; a perfect matching is a matching that includes edges touching
Edge_coloring
Partition of the vertices of a graph
left out, there is a perfect matching of the remaining vertices. In particular, each component has a near-perfect matching: a matching that covers all but
Gallai–Edmonds_decomposition
Graph theory problem: find a matching containing the most edges
a maximal matching is also the largest non-overlapping cover of the graph. If all the vertices are covered, we call it a perfect matching. For finite
Maximum-cardinality_matching
Bipartite graph partition with special property
the same subset if and only if they are paired with each other in a perfect matching of the graph. It is named after A. L. Dulmage and Nathan Mendelsohn
Dulmage–Mendelsohn decomposition
Dulmage–Mendelsohn_decomposition
polynomial time by transforming the problem into a problem of finding a perfect matching in a larger graph. If the cycles of the cover have no edges in common
Vertex_cycle_cover
Undirected cubic graph with 12 vertices and 18 edges
covered by four perfect matchings. This property plays a key role in a proof that testing whether a graph can be covered by four perfect matchings is NP-complete
Tietze's_graph
Graphs formed by a hypercube's edges and vertices
copies of Q n − 1 {\displaystyle Q_{n-1}} connected to each other by a perfect matching. Hypercube graphs should not be confused with cubic graphs, which are
Hypercube_graph
Assignment problem in combinatorial mathematics
Hamiltonian cycles in crown graphs. A crown graph is formed by removing a perfect matching from a complete bipartite graph Kn,n; it has 2n vertices of two colors
Ménage_problem
Undirected graph derived from a hypercube graph
an undirected graph formed from a hypercube graph by adding to it a perfect matching that connects opposite pairs of hypercube vertices. The folded cube
Folded_cube_graph
Computation complexity problem
{\displaystyle {\mathcal {M}}_{n}} denotes the family of all possible perfect matchings on n {\displaystyle n} nodes). Their goal is to output a tuple ⟨ i
Hidden_matching_problem
Natural number
other number with this property (in decimal) is 27. There are 15 perfect matchings of the complete graph K6 and 15 rooted binary trees with four labeled
15_(number)
Mathematical optimization problem
weight bipartite matching problem or assignment problem is to find a perfect matching M ⊆ E whose total weight is minimized. The idea is to reduce this problem
Minimum-cost_flow_problem
Complexity class
sortings. A single perfect matching can be found in polynomial time, but counting all perfect matchings is #P-complete. The perfect matching counting problem
♯P-complete
Decision problem in computer science
every zone, that is, (20+21+...+23n-1). If the 3DM instance has a perfect matching, then summing the corresponding integers in the SSP instance yields
Subset_sum_problem
Block design in combinatorial mathematics
vertices, 15 edges, 15 perfect matchings, and 6 different 1-factorizations (ways to partition the edges into disjoint perfect matchings). The set of vertices
Steiner_system
Graph without four-vertex star subgraphs
them: the fact that all claw-free connected graphs of even order have perfect matchings, the discovery of polynomial time algorithms for finding maximum independent
Claw-free_graph
Polytope
assignment polytope, the polytope of doubly stochastic matrices, or the perfect matching polytope of the complete bipartite graph K n , n {\displaystyle K_{n
Birkhoff_polytope
Characterization of the size of a maximum matching in a graph
characterization of the size of a maximum matching in a graph. It is a generalization of Tutte's theorem on perfect matchings, and is named after W. T. Tutte (who
Tutte–Berge_formula
Graph with all vertices of degree 3
graph has a perfect matching. Lovász and Plummer conjectured that every cubic bridgeless graph has an exponential number of perfect matchings. The conjecture
Cubic_graph
Theorem that deals with the decompositions of complete hypergraphs
{n}{2}}{\frac {2}{n}}=n-1} colors so that the edges of each color form a perfect matching. Baranyai's theorem says that we can do this whenever n {\displaystyle
Baranyai's_theorem
Four-dimensional analogue of the cube
counted by mapping the nets to paired trees (a tree together with a perfect matching in its complement). One of these unfoldings is the Dali cross, named
Tesseract
Adjusting input/output impedances of an electrical circuit for some purpose
In electrical engineering, impedance matching is the practice of designing or adjusting the input impedance or output impedance of an electrical device
Impedance_matching
decomposition of a complete graph of even order minus a 1-factor (a perfect matching) into even cycles and a complete graph of odd order into odd cycles
Cycle decomposition (graph theory)
Cycle_decomposition_(graph_theory)
Partition of a graph whose components are reachable from all vertices
bipartite graph, according to whether or not they can be part of a perfect matching in the graph. A directed graph is strongly connected if and only if
Strongly_connected_component
Topics referred to by the same term
theory), a property describing how far a given graph is from having a perfect matching Deficiency (medicine), including various types of malnutrition, as
Deficiency
Problem in linear algebra
number of perfect matchings in a graph. For planar graphs (regardless of bipartiteness), the FKT algorithm computes the number of perfect matchings in polynomial
Computing_the_permanent
Bipartite graph where each node of 1st set is linked to all nodes of 2nd set
complete bipartite subgraphs Crown graph, a graph formed by removing a perfect matching from a complete bipartite graph Complete multipartite graph, a generalization
Complete_bipartite_graph
Graph with all vertices of degree 4
bipartite quartic graph has a perfect matching. In this case, a much simpler and faster algorithm for finding such a matching is possible than for irregular
Quartic_graph
admits a perfect matching if and only if the polynomial det(A) in the xij is not identically zero. Furthermore, the number of perfect matchings is equal
Edmonds_matrix
Graph with equal-size maximal independent sets
in a very well covered graph, and then use a matching algorithm to test whether H has a perfect matching. Some problems that are NP-complete for arbitrary
Well-covered_graph
Ancient sculptures in Sardinia (Italy)
archaeological sites after the collapse of top parts – confirm the perfect matching of these models with the Nuragic architecture of Middle and Recent
Giants_of_Mont'e_Prama
On domino tiling after removing two corners
Joseph E.; Heule, Marijn J. H.; Bryant, Randal E. (2021), "Bipartite perfect matching benchmarks" (PDF), in Balyo, Tomáš; Froleyks, Nils; Heule, Marijn;
Mutilated_chessboard_problem
Solution to x*x + y*y + z*z = 3xyz
Uniqueness Conjecture: A Mathematical Journey from Irrational Numbers to Perfect Matchings. Cham Heidelberg: Springer. ISBN 978-3-319-00887-5. MR 3098784. Cassels
Markov_number
Type of graph in mathematics
the same as in the half-graph itself. The half graph has a unique perfect matching. This is straightforward to see by induction: u n {\displaystyle u_{n}}
Half_graph
3-regular graph with 30 vertices and 45 edges
corresponds to an edge or a perfect matching, and connected vertices represent the incidence structure between edges and matchings. Based on this construction
Tutte–Coxeter_graph
Type of graph vertex labeling
graphs and caterpillar graphs are graceful. All lobster graphs with a perfect matching are graceful. All trees with at most 27 vertices are graceful; this
Graceful_labeling
Counting technique in combinatorics
construction of the chromatic polynomial of a graph. The number of perfect matchings of a bipartite graph can be calculated using the principle. Given
Inclusion–exclusion_principle
Graph where every edge is in one triangle
point of view of any one vertex) the rest of the graph looks like a perfect matching. Locally linear graphs have also been called locally matched graphs
Locally_linear_graph
1972 single by Jim Croce
the Chuck Berry 'Memphis' vein" and said that "the single is a near-perfect matching of this singer to the song." In 1973, Croce performed "Operator (That's
Operator (That's Not the Way It Feels)
Operator_(That's_Not_the_Way_It_Feels)
Topics referred to by the same term
theory), a subgraph in which removing any vertex leaves a graph with a perfect matching HMS Blossom, three Royal Navy ships Operation Priha (Blossom), a series
Blossom_(disambiguation)
German reality television series
confirms a perfect match, that couple will go to the honeymoon suite and will automatically be paired up for the remainder of the matching nights. At
Are You the One? (German TV series)
Are_You_the_One?_(German_TV_series)
Mathematical proof about the permanent of matrices
each, the number of perfect matchings equals the permanent of its biadjacency matrix and the square of the number of perfect matchings is equal to the permanent
♯P-completeness of 01-permanent
♯P-completeness_of_01-permanent
Non-crossing graph with vertices on outer face
the problem of determining the planarity of graphs formed by using a perfect matching to connect two copies of a base graph (for instance, many of the generalized
Outerplanar_graph
Aspect of mathematical group theory
has 15 edges, which can be partitioned into perfect matchings in 15 different ways, each perfect matching being a set of three edges no two of which share
Automorphisms of the symmetric and alternating groups
Automorphisms_of_the_symmetric_and_alternating_groups
Technique for reducing number of solutions
{\mathcal {F}}} is the set of perfect matchings, so that with probability at least 1/2, there exists a unique perfect matching. When each indeterminate x i j
Isolation_lemma
Theorem in graph theory
′ {\displaystyle G'} can be partitioned into k {\displaystyle k} perfect matchings by a theorem of Kőnig. Now merging v ′ {\displaystyle v'} with v ″
2-factor_theorem
Song by Kris Kristofferson and Fred Foster
recording of it was the day after she died. Record World called it a "perfect matching of performer and material". Joplin's version topped the charts to become
Me_and_Bobby_McGee
Family of graphs with 2n nodes and n(n-1) edges
be viewed as a complete bipartite graph from which the edges of a perfect matching have been removed, as the bipartite double cover of a complete graph
Crown_graph
Strongly NP-complete problem in computer science
(T/3,T/5). Given a perfect matching in E, we construct a 4-partition of ABCD as follows: For each triplet t= {wi,xj,yk} in the matching, we construct a 4-set
3-partition_problem
Set that intersects every one of a family of sets
(defined as a system of distinct representatives) is equivalent to a perfect matching in this graph. One can construct a hypergraph in which the vertices
Transversal_(combinatorics)
Combinitorics of Polyhedra
on this polytope can be interpreted as a bipartite minimum weight perfect matching problem. The Birkhoff–von Neumann theorem states that this polytope
Polyhedral_combinatorics
Subunit of a computational problem
the problem of finding a subgraph with given degree constraints to a perfect matching problem. However, the "gadget" terminology has a later origin, and
Gadget_(computer_science)
Upper bound on intersecting set families
the perfect matchings of a complete bipartite graph K n , n {\displaystyle K_{n,n}} and the theorem states that, among families of perfect matchings each
Erdős–Ko–Rado_theorem
Undirected graph with 14 vertices
distance regular. There are 24 perfect matchings in the Heawood graph; for each matching, the set of edges not in the matching forms a Hamiltonian cycle.
Heawood_graph
Cyclic order and one-to-one pairing of a set of objects
cyclic order on a set of objects, together with a one-to-one pairing (perfect matching) of those objects. Chord diagrams are conventionally visualized by
Chord_diagram_(mathematics)
Graph formed by subdivision of triangles
one perfect matching. However, in this case more is known: the duals of Apollonian networks always have an exponential number of perfect matchings. László
Apollonian_network
graph G = (V, E) is a matrix used to determine the existence of a perfect matching: that is, a set of edges which is incident with each vertex exactly
Tutte_matrix
Least-weight tree connecting graph vertices
maximum flow problem), and approximating the minimum-cost weighted perfect matching. Other practical applications based on minimal spanning trees include:
Minimum_spanning_tree
Topic in algebraic graph theory
admit perfect state transfer at time t {\displaystyle t} . Moreover, a graph G {\displaystyle G} must have a perfect matching that admits perfect state
Continuous-time_quantum_walk
Finding shortest walks through all graph edges
graph, and then finding a minimum weight perfect matching in this complete graph. The edges of this matching represent paths in the original graph, whose
Chinese_postman_problem
Decomposition of a graph into hamiltonion cycles
decomposition of the complete k {\displaystyle k} -uniform hypergraph into perfect matchings. Every 4-regular undirected graph has an even number of Hamiltonian
Hamiltonian_decomposition
Czech mathematician and computer scientist
graph has an exponential number of perfect matchings, strengthening Petersen's theorem that at least one perfect matching exists. In a pair of papers with
Daniel_Kráľ
Function of a matrix
Thus, just as the hafnian counts perfect matchings in a graph from its adjacency matrix, the permanent counts matchings in a bipartite graph from its biadjacency
Hafnian
Desk lamp & mascot of Pixar Animation Studios
light-source, moving around and self-shadowing the world around him, was a perfect matching of technology and subject matter." Luxo Jr. made its debut at the 1986
Luxo_Jr._(character)
American rock supergroup
A Perfect Circle is an American rock supergroup formed in Los Angeles, California, in 1999 by guitarist Billy Howerdel and Tool vocalist Maynard James
A_Perfect_Circle
satisfies a generalization of Hall's marriage theorem: it admits a perfect matching iff for all disjoint vertex-sets V1, V2, if | e ∩ V 2 | ≥ | e ∩ V 1
Balanced_hypergraph
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