# Questions tagged [perfect-matchings]

A perfect matching is a matching of all the vertices of a graph. In other words, a perfect matching is a set of edges such that each vertex of the graph is incident to exactly one edge in the set.

63 questions

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### Minimum planar bipartite graph to cover all perfect matching count

Given set $\mathcal T_n=\{0,1,\dots,2^n-1\}$ what is the minimum number of vertices $2m$ needed in a planar bipartite balanced graph such that at every $i\in\mathcal T_n$ there is a graph $G\in\...

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+100

### Vertex Coloring inherited from Perfect Matchings (motivated by Quantum Physics)

The following purely graph-theoretic question is motivated by quantum mechanics (and a special case of the questions asked here).
Bi-Colored Graph: A bi-colored weighted graph $G=(V(G),E(G))$, on $n$ ...

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### Triangle Center from Weighted Perfect Matchings

let $\Delta$ be the triangle whose corners $A$, $B$, $C$ points in general position in Euclidean plane and, let $D$ be a fourth point inside $\Delta$.
Question:
what is known about the ...

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### Name for Spanning Trees Containing all Edges of a Minimum Weight Perfect Matching

This question is motivated by the task of "uniformly" bicoloring the vertices of a symmetric TSP-instance graph with $2n$ vertices.
A simple heuristical requirement for such a bicoloring could be ...

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### Why is the number of Perfect Matchings in a triangular grid equivalent to the number of Royal Paths?

The sequence A006318 at OEIS stands for the Schröder numbers.
They describes the number of lattice paths from the southwest corner $(0,0)$ of an $n\times n$ grid to the northeast corner $(n,n)$, ...

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### Can Orientability of Manifolds be Generalized to TSP Instances?

It is well known, that there are two basic kinds of manifolds, orientable and non-orientable ones; the most simple examples being obtained by identifying a pair of opposite sides of a rectangular ...

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### Complexity of Calculating Minimum Weight Final Perfect Matchings

It is known, that Minimum Weight Perfect Matching can be calculated in $O(n^3)$;
Furthermore, it is possible, that the edge sets of the Minimum Weight Perfect Matching and of the Maximum Weight ...

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### Why is the number of Hamiltonian Cycles of n-octahedron equivalent to the number of Perfect Matching in specific family of Graphs?

In OEIS A003436, it is written that the number of inequivalent labeled Hamilton Cycles of an n-dimesnional Octahedron is the same as the number of Perfect Matchings in a the complement of the Cycle ...

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### Graphs with only disjoint perfect matchings, with coloring

The following purely graph-theoretic question is motivated by quantum mechanics.
Definitions: A bi-colored graph $G$ is an undirected graph where every edge is colored. An edge can either be ...

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### Infinite graphs with large degree but no perfect matching [duplicate]

Is there an example of an infinite connected, simple, undirected graph $G = (V,E)$ such that every vertex has $|V|$ neighbors, but $G$ does not have a perfect matching (that is, a set $M\subseteq E$ ...

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### How to characterize “matching-transitive” regular graphs?

I am interested in regular graphs $G$ such that for each pair of 1-factors (=perfect matchings) $F$ and $F'$ there is an automorphism of $G$ that takes $F$ to $F'$. Let's call this property matching ...

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### Complexity for calculating number of Perfect Matchings in k-regular hypergraph

Let $G(V,E)$ be a unweighted, k-regular hypergraph, with vertices $V=(v_1, ... v_n)$ and edges $E=(e_1, ... e_m)$. The k-regularity leads to $|e_i|=k$ (i.e. every edge contains exactly $k$ vertices).
...

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### Matching and minimal degree

Let $n\in\mathbb{N}$ be a positive integer and let $G =(V,E)$ be a connected simple undirected graph with $|V| = 2n$. Is it true that if for the minimal degree $\delta(G)$ we have $\delta(G) \geq n$, ...

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### Connected infinite graph $G$ with $\delta(G)\geq 2$ and no perfect matching [closed]

Is there a connected infinite graph $G=(V,E)$ such that $\text{deg}(v) \geq 2$ for all $v\in V$, and $G$ possesses no perfect matching?

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### Perfect matchings in infinite graphs

Let $G=(V,E)$ be an infinite graph such that $|V| = \kappa$ for some infinite cardinal $\kappa$, and every $v\in V$ has degree $\kappa$. Does $G$ have a perfect matching?

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### Does the Hadwiger-Nelson graph have a perfect matching?

The Hadwiger-Nelson graph on $\mathbb{R}^n$ is defined to be $(\mathbb{R}^n,E_n)$ where $$E_n = \big\{\{x,y\}: x,y\in \mathbb{R}^n \text{ and } |x-y|=1\big\},$$ where $|\cdot|$ denotes the Euclidean ...

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### Is every positive integer the permanent of some 0-1 matrix?

In the course of discussing another MO question we realized that we did not know the answer to a more basic question, namely:
Is it true that for every positive integer $k$ there exists a balanced ...

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323 views

### Graph to Bipartite conversion preserving number of perfect matchings

Given a graph $G$ on $n$ vertices is there a technique to convert to a balanced bipartite graph $B$ with $O(n^c)$ vertices at some fixed $0<c$ in $O(n^{c'})$ time at some fixed $0<c'$ such that ...

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### Applications of Perfect Matching

I'm exploring some applications of perfect matching and I would like some input. I have found many applications in chemistry (storing information, estimating bond lengths, estimating resonance energy, ...

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### Maximum number of perfect matchings in a planar graph?

What is the maximum number of perfect matchings a planar $k$-partite $|V|$ number of vertices simple graph can have where $k=2,3,4$ ($k>4$ is impossible for a planar graph)?
Since number of ...

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55 views

### Minimum size of genus $g$ bipartite graphs with $2^n$ perfect matchings

Given $n\in\Bbb Z_{\geq0}$ let $2T_{n,g}$ be size of smallest number of vertices of genus $g$ bipartite graph with $T_{n,g}$ vertices of each color such that number of perfect matchings is $2^n$.
Eg: ...

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### Applications of Hafnians

I am learning about Hafnians but I am struggling to find real-world applications of them. I understand the applications of determinants, permanents, and even pfaffians but I am at a loss for Hafnians. ...

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### Fastest algorithm for counting perfect matchings in a general graph

Let $G(V,E)$ be a undirected graph. I am interested in the fastest known algorithm for counting the number of perfect matchings in $G(V,E)$ (which is known to be in $\#P$). In particular, what is the ...

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### Unique matching completion

Assume we have bridgeless cubic graph $G(V, E)$, $n=|V|$.
By Petersen's theorem, every such graph has a perfect matching.
Moreover, given any edge in $G$ there exists a perfect matching containing ...

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### Graphs with only disjoint perfect matchings

Let $G(V,E)$ be a graph. I am searching for graphs with only disjoint perfect matchings (i.e. every edge only appears in at most one of the perfect matchings).
Examples:
Cyclic graph $C_n$ with even ...

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555 views

### Edmond's blossom algorithm for Max weight perfect matchings

Edmond's blossom algorithm computes a maximum weight matching in a general graph (https://en.wikipedia.org/wiki/Blossom_algorithm).
Many papers also reference to Edmond's blossom algorithm to compute ...

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695 views

### Number of perfect matchings in bipartite graph with given minimum degree

Let $G$ be a spanning subgraph of $K_{n,n}$ with minimum degree $\delta(G) \geq n/2$. It's easy to show using Hall's theorem that $G$ has a perfect matching, and the example of two disjoint copies of ...

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### Reference Request: “Resolutions” of $K_n$ for $n$ odd

A resolution (in the combinatorial design sense) of $K_{n}$ is a collection of sets of edges of $K_{n}$ so that within each set of edges, each vertex appears once, and over the entire collection, each ...

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### Missing count in number of perfect matchings

Let $f(G)$ give number of perfect matchings of a graph $G$.
Denote $\mathcal N_{2n}=\{0,1,2,\dots,n!-1,n!\}$.
Denote collection of all $2n$ vertex balanced bipartite graph to be $\mathcal G_{2n}$.
...

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108 views

### On Schrijver's lower bound for the number of perfect matchings

Schrijver's lower bound gives the number of perfect matchings in a $k$-regular bipartite graph as $\Big(\frac{(k-1)^{k-1}}{k^{k-2}}\Big)^n$. What is the corresponding lower bound for minimum-degree $k$...

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### Expected number of perfect matchings in bounded degree bipartite graphs

Consider collection $\mathcal C_{n,n,\Delta}$ of every $2n$ vertex balanced bipartite graph of average degree $\Delta$.
What is the expected number of perfect matching a graph in $\mathcal C_{n,n,\...

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348 views

### On number of perfect matchings

Consider $2n$ vertex balanced bipartite graph.
If total number of edges is $n^2$ then we have $n!$ perfect matchings.
Fix $c\in(0,\frac12)$ and consider collection of $2n$ vertex balanced bipartite ...

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### Largest number of perfect matchings in bounded genus graphs

What is the largest number of perfect matchings a genus $g$ bipartite graph on $n+m$ vertices have?

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### Fraction of graphs with bound on number of perfect matchings

Asymptotically what is the fraction of balanced bipartite graph on $2n$ vertices with at most $cn^{\beta}$ edges having at most $n^\alpha$ perfect matchings for any fixed $c,\alpha>0$ and fixed $\...

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### On symmetric difference of $k$-partite perfect matchings

Given a bipartite graph we know that symmetric difference of any two perfect matchings is union of even cycles.
Conversely when is it true that every union of even cycles comes from symmetric ...

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### Probability bound for perfect matching

Let $p<1$ be a constant. Consider two sets $A,B$, each with $n$ vertices. For each pair $(a,b)\in A\times B$, the edge between $a$ and $b$ appears with probability $p$, independently of the ...

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### Does every 3-regular bridgeless graph have a perfect matching? [closed]

Let $G$ be a simple $3$-regular (every vertex has degree $3$) $2$-edge connected graph. Does $G$ contain a perfect matching?

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### “Hypo” and “Hyper” for Perfect Matching

There is a fairly rich classification on graphs with respect to the existence of Hamiltonian cycles either in unmodified graphs or after certain small modifications.
Do there also exist such ...

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### Densest Graphs with Unique Perfect Matching

Given a graph $G$ with $n$ vertices, that has a perfect matching $M$, what is the maximal number of edges that $G$ can have without contradicting the uniqueness of $M$?
Are examples of such extremal ...

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### Test Instances for Perfect Matchings in Graphs

Are there any graphs with a known set of perfect matchings and other predefined properties, such as vertex connectivity, which can be used for testing the implementation of matching algorithms?
...

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### Algorithm to count the number of perfect matchings in non planar graph

I need to count the number of perfect matchings of a certain family of graphs. This family of graph is non planar and a type of snark. For the initial cases, it seems that this number is growing ...

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### The cost function in the Weighted Bipartite Matching Problem (a.k.a the Assignment Problem)

In the definition of this problem, the weight/cost function generally takes value in $\mathbb{Z}$ (or sometimes $\mathbb{Q}$).
This is what I observed from some books (e.g. "Combinatorial ...

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### Matching with probabilistic edges

Let $p<1$ be a constant. Consider two sets $A,B$, each with $n$ vertices. For each pair $(a,b)\in A\times B$, the edge between $a$ and $b$ appears with probability $p$, independently of the ...

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### About structure of the set of perfect matchings of $K_{n,n}$

Are there any special properties known about the set of perfect matchings of $K_{n,n}$? Like any global structure of this set? Some natural way to partition it? Like is there some algebraic structure ...

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### Minimum number of perfect matchings in a regular bipartite graph

Is there a lower bound on the number of perfect matchings in a $k$-regular bipartite graph?
One can use Hall's marriage theorem and induction on $k$ to derive the lower bound of $k$. I can't come up ...

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### Postnikov's approach to perfect matchings of graphs

Over a decade ago Alexander Postnikov developed his own way of looking at perfect matchings of bipartite plane graphs. As I recall, he starts with a 2-coloring of the square grid and creates a new ...

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### Mixing time for dimers on the square-octagon graph

Consider the "fortress graph" of order $n$ (see Figure 9 of http://faculty.uml.edu/jpropp/tiling/www/mdblum/arctic.html). It's been known empirically for twenty years that if one turns the set of ...

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### Counting perfect matchings with integrals

Has anyone used the Joni-Rota-Godsil integral formula (see details below) to count perfect matchings of square-grid graphs, Aztec diamond graphs, hexagon-honeycomb graphs, etc.? (Or even just to ...

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### Kasteleyn, Gessel-Viennot and eigenvalues

The Kasteleyn matrix (for counting perfect matchings) and the Lindström-Gessel-Viennot matrix (for counting families of nonintersecting lattice paths) are tightly related, as observed many times by ...

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### Complexity of finding three perfect matchings with no edge in common in a bridgeless cubic graph

According to a conjecture:
Conjecture (Fan & Raspaud, 1994) Every bridgeless cubic graph contains three perfect matchings with no edge in common.
Equivalent statement here
Main question:
...