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...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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....

S\cup T}y(v)} . The cost of each perfect matching is at least the value of each potential: the total cost of the matching is the sum of costs of all edges...

formed from complete bipartite graphs by removing the edges of a perfect matching. Hypercube graphs, partial cubes, and median graphs are bipartite....

assignment, and the graph-theoretic version is called minimum-cost perfect matching. Otherwise, it is called unbalanced assignment. If the total cost of...

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...

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...

graph theory, a 3-dimensional matching is a generalization of bipartite matching (also known as 2-dimensional matching) to 3-partite hypergraphs, which...

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...

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...

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...

{\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...

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...

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...

sortings. A single perfect matching can be found in polynomial time, but counting all perfect matchings is #P-complete. The perfect matching counting problem...

and graph theory, the maximum weight matching problem is the problem of finding, in a weighted graph, a matching in which the sum of weights is maximized...

decomposed into two copies of Qn − 1 connected to each other by a perfect matching. Hypercube graphs should not be confused with cubic graphs, which are...

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...

a perfect matching. Birkhoff's algorithm is a greedy algorithm: it greedily finds perfect matchings and removes them from the fractional matching. It...

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...

{\displaystyle K_{5}} minor can be formed by contracting the edges of a perfect matching, for instance the five short edges in the first picture. The K 3 ,...