# 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.

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### Generalizations of a theorem of Edmonds/Tutte on existence of a perfect matching in a graphs

It is well known that for a bipartite graph $G$ with bi-adjacency matrix $A$, then $\det A \neq 0$ (as a polynomial) iff $G$ has a perfect matching (there is a similar result for general graphs with ...

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### Algorithms to count perfect matchings in near planar graphs

It is well known that counting perfect matchings is tractable in planar graphs (due to Kastelyn).
I am interested in classes of (for lack of a better word) "near" planar graphs (1-planar, ...

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### Detecting non-optimality in disjoint unions of perfect matchings

This is a follow-up question to Minimum-weight disjoint union of perfect matchings:
let $G$ be a complete symmetric graph with $2n$ vertices, whose edges are mapped to their weights by $\omega()$ and ...

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### Generalizing Hall's marriage theorem to non-perfect matchings

Let $G = (X, Y, E)$ be bipartite graph such that $|X|=|Y|=n$.
A matching $M \subseteq E$ is a subset of disjoint edges
(i.e., there does not exist a pair of edges $(x, y) \in M$ and $(x', y') \in M$ ...

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### Optimal perfect matchings in magic squares

Question:
what is known about minimum/maximum weight perfect matchings in magic squares with or without special properties like e.g. being pandiagonal?
I am especially interested minimal/maximal ...

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### Calculation of cardinality constrained minimum weight matchings

Given a complete weighted graph $G(V,E),\ |V|=2n$, calculating a minimum weight matching with $n-k$ edges can be reduced to calculating a perfect matching in $H(V+U,E+F),\ |U|=2k,\ F=(u\in U,v\in V),\ ...

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### Pair matching between divisors less and more than $\sqrt{N}$

Let $n$ be the positive integer. Let $A$ and $B$ be sets of divisors of $n$ less and more than $\sqrt{n}$ respectively.
Consider bipartite graph $(A, B)$, where two vertices are connected when one ...

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### How to understand Chegireddy-Hamacher's algorithm for finding k-best perfect matching

I am reading Algorithms for finding K-best perfect matchings by Chegireddy and Hamacher, and I have trouble to understand their Section 2 "General algorithm for K-best perfect matchings ". ...

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### Are there decompositions of $K_{16}$ by certain 3-regular graphs?

This is inspired by the problem of the Hoffman-Singleton Decomposition of $K_{50}$. I wanted to look at smaller variants of this kind of problem, and so naturally I started wondering:
Can the (edges ...

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### Reducing minimum-weight $f$-subfactors to minimum-weight perfect matching

Question:
What is known about reducing the calculation of minimum-weight $f$-subfactors of symmetric graphs in the presence of negative edge weights to minimum-weight perfect matching?
By a $f$-...

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### Finding minimum weight perfect matchings for the same graph with slightly different edge weights

Suppose I have found the minimum weight perfect matching (MWPM) for a given weighted graph, with say, the Blossom algorithm. Now if the weights of a small subset of edges are changed in the graph, how ...

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### Counting number of perfect matchings

Counting perfect matchings in bipartite graphs is $\# P$ complete. Let $G(V,E)$ be a graph known to have $d$ number of perfect matchings. Bipartite it the obvious way by adding $E$ vertices with one ...

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### Existence of certain regular graphs

Question:
what can be said about the existence of $2k$ regular graphs, $1\lt k$ that have a $1$-factor and a $2$-factor?
Provided their existence, what is/are the smallest for $k$?
The graphs must be ...

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### Counting matchings and perfect matchings

A matching in a graph is a subset of the edges such that
no two edges share a vertex. A perfect matching is a matching where every vertex is part of exactly one edge in the matching.
Counting the ...

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### Finding a bipartite graph that contains a specific elements of perfect matchings

I am a physicist who is interested in the applications of graph theory.
I've been studying the bipartite graphs and perfect matching finding problems. I see there are several research works on ...

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### An "incomplete" tiling?

Given an $m\times n$ chess board, we place $p$ $2\times 1$ dominoes on the board so that they don't overlap. How many ways can we place them?
When each square of the board is covered by a domino this ...

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### Bounds for smallest non-trivial designs

Given $s>t\ge 2$, let $N(s,t)$ be the smallest integer $n>s$ such that there exists an “$(n;s;t;1)$-design” (i.e., a collection of $s$-subsets $e_1,\dots,e_m$ of $[n]:=\{1,\dots,n\}$, such that ...

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### Edge-length constraints from greedy matching

The subject of this question are perfect matchings of a complete undirected graph $G(V,E), n:=\mathrm{card}(V)=2k$, without self-loops or parallel edges and $n=2k$ vertices.
The objective is to ...

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### Hopcroft–Karp Algorithm for a dynamic graph

As so you all know, we have Hopcroft–Karp Algorithm for maximum matching between two sides in a bipartite graph. It runs in $O(\sqrt{V} \times E)$ where $V$ is the vertex set and $E$ is the edges set.
...

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### Curious identity involving the number of perfect matchings of the complete graph

Can you prove (preferably combinatorially) the following identity for the total number of perfect matchings of the complete graph $K_{2n}$, where the edges in the matching are ordered, i.e., $\binom{...

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### Bound on the number of maximum matchings in a graph

It is known that the number of perfect matchings in a graph is bounded above by the integer part of the square root of the permanent of its adjacency matrix. But, suppose I take the square root of the ...

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### Symmetry of optimal solutions to symmetric assignment problems

Is there a sound proof of or a counter example to the following conjecture:
if $\boldsymbol{A}^T=\boldsymbol{A}$ is the cost-matrix of a bipartite assignment problem with unique optimal assignment,
...

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### Connecting $2n$ points in $\mathbb R^2$ with line segments s.t. each point belongs to exactly one line segment

I'm trying to do a certain simulation related to the toric code and I'm looking for an algorithm that connects $2n$ points ($n \in \mathbb Z_+$) in $\mathbb R^2$ with line segments with the following ...

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### Upper bound on the number of perfect matchings in $K_{3,3}$-free graphs

Let $G=(V,E)$ be a graph with an even number of vertices $|V|=2n$. Assume that $G$ is $K_{3,3}$-free i.e. it does not contain a graph isomorphic to biclique $K_{3,3}$. A perfect matching of $G$ is a ...

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### On perfect matchings on planar graphs - is there a linear time deterministic algorithm?

The slides here provide a way to get a pfaffian orientation from Minimum Spanning Tree.
MST can be found in linear time if graph is planar and weights are $1$ and the slides give a linear time ...

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### What's the name of the graph operation of connecting two copies of a graph with a perfect matching?

Let $G=(V_1,E_1)$ be a simple graph with vertex set $\{v_1,v_2,\ldots,v_n\}$ and let $G'=(V_2,E_2)$ be another copy of $G$ with vertex set $\{u_1,u_2,\ldots,u_n\}$. Assume $V_1\cap V_2= \emptyset$.
...

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### The perfect matching problem of planar graph

We know that connectivity is closely related to the Hamiltonian of planar graphs.
The most famous result is the Tutte theorem.
Theorem (Tutte, 1956). A 4-connected planar graph has a Hamiltonian ...

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### Sum of number of perfect matchings and a constant constuction

Suppose we have two bipartite graphs $G_1$ and $G_2$ with perfect matching count $P_1$ and $P_2$ respectively then their disjoint union gives a bipartite graph with perfect matching $P_1P_2$.
Is ...

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### Disjoint perfect matchings in complete bipartite graph

Let $K_{n,n}$ be a complete bipartite graph with two parts $\{u_1,u_2,\ldots,u_n\}$ and $\{v_1,v_2,\ldots,v_n\}$, and let $K^-_{n,n}$ be the graph derived from $K_{n,n}$ by delete a perfect matching $\...

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### Generalization of Marshall Hall's Theorem to non-simple bipartite graphs

Lemma 8.6.5 of the book "Matching Theory" by Lovász and Plummer states the following lemma:
Lemma: Let $G$ be a simple bipartite graph with bipartition $(A,B)$, and assume that each point ...

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### Berge-Fulkerson conjecture --- the planar case

A well-known conjecture of Berge and Fulkerson says that every bridgeless cubic graph has a collection of six perfect matchings that together cover every edge exactly twice. Is this still open for ...

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### Sum of perfect matching construction

Suppose we have two bipartite graphs $G_1$ and $G_2$ with perfect matching count $P_1$ and $P_2$ respectively then their disjoint union gives a bipartite graph with perfect matching $P_1P_2$.
Is ...

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### Succinct polynomial sized representation of balanced bipartite graphs whose perfect matching count is a primorial

Is there a $P$ time definable sequence of succinct polynomial sized representation of balanced bipartite graphs whose number of perfect matchings is a primorial?
For factorial a complete bipartite ...

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### Bipartite graph with exactly one perfect matching

$\textbf{Problem:}$ Find all bipartite graphs $G[X,Y]$ satisfying the following properties:
$1.$ $|X|=|Y|$, where $|X|\ge 2$ and $|Y|\ge 2$.
$2.$ All vertices have degree three except for two vertices ...

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### A rainbow perfect matching in an edge-colored graph with spanning color classes

This question is a sequel of my last question and is eventually motivated by recent advances in quantum physics. Given an even number $n\ge 6$ and a positive integer $k<n$, Claim from the linked ...

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### At most one perfect matching of a bipartite graph

I. Given biadjacency matrix $A$ of a bipartite graph on $2n$ vertices having $n$ vertices of either color on the constraints the graph either has
$0$ perfect matchings
$1$ perfect matchings
is it ...

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### Why is Schröder numbers equivalent to the number of perfect matchings for triangular grid of n squares and how the graph look like? [duplicate]

In the OEIS entry for the Schröder numbers is A006318. There is a comment which related the sequence to perfect matchings:
The number of perfect matchings in a triangular grid of n squares (n = 1, 4, ...

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### Dyadic distribution of $0/1$ permanents

Fix reals $a,b\in(1,2)$ satisfying $1<b<a<ab<2$.
What fraction of $0/1$ matrices of dimensions $n\times n$ have permanents
in $[b2^m,a2^m]$ at some $m\in\{0,1,2,\dots,\lfloor\log_2n!\...

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### Computing bipartite matching of size $k$?

Given a bipartite graph with $n$ vertices on each side and an integer $k$, how can we compute all bipartite matchings of size $k$?
The problem of computing all perfect matchings is #P-complete. But I ...

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### Minimum-weight disjoint union of perfect matchings

Is there a counter example or proof for the claim that the lightest edge-disjoint union of a pair of perfect matchings contains the edges of the lightest perfect matching in a finite complete graph ...

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### Smallest $3$-regular graph with a unique perfect matching

What is the smallest 3-regular graph to have a unique perfect matching?
With a large enough number of nodes, it is possible for a 3-regular graph to have no perfect matching (example can be seen in ...

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### Number of extremal $\{0,1\}$ matrices having permanent $1$ property

Is there a function which describes the number of $\{0,1\}^{n\times n}\cap\mathbb Z^{n\times n}$ matrices having permanent $1$?
I think it might be $\mathsf{poly}(n!)$ bounded.
Is there a function ...

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### Determining a specific perfect matching $M$ by repeatedly asking for $|M \cap M_i|$ for other perfect matchings $M_i$

Let $G=(V,E)$ be a complete bipartite graph with $2n$ vertices and $M \subset E$ some unknown perfect matching of $G$.
The goal is to determine $M$ by repeatedly choosing some perfect matching $M_i \...

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### Unique bipartite perfect matchings and cycles?

Given a graph $G$ which is bipartite and balanced and has unique perfect matching let $G^{e}$ be $G$ without edge $e$. Let $G\cup G_{\pi,\pi'}$ be union of $G$ and $G_{\pi,\pi'}$ where $G_{\pi,\pi'}$ ...

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### Spanning subgraphs defined via $K_4$ matchings

I have by accident found an interesting kind of spanner of complete symmetric graphs $G(V,E)$ with weighted edges.
What I actually had planned was to implement an algorithm for calculating certain non-...

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### Density of perfect matching count in $k$-partite graphs?

Let $f(G)$ give number of perfect matchings of a graph $G$.
Denote $\mathcal N_{n}=\{0,1,2,\dots,n!-1,n!\}$.
Denote collection of all $kn$ vertex balanced $k$-partite graph (each color is on $n$ ...

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### Is there a bipartite graph whose determinant corresponds to number of perfect matchings?

Let $M\in\{0,1\}^{n\times n}$ be a square integer matrix. If we consider $M$ as biadjacency of a balanced bipartite graph on $2n$ vertices having $n$ vertices of color $1$ and $n$ vertices of color $2$...

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### Mod $2$ information on perfect matchings in general graphs

Determinant modulo $2$ of biadjacency matrix of bipartite graphs provide mod $2$ information on number of perfect matchings on bipartite graphs providing polynomial complexity in bipartite situations.
...

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### Perfect matchings in infinite regular bipartite graphs

This question was motivated by a discussion here and is related to a previous question here.
Let $\kappa$ and $\lambda$ be cardinals such that $0<\lambda\leq \kappa$. Let $G=(A\cup B, E)$ be a ...

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### Infection on a complete graph

Suppose we have a complete graph on $2n$ vertices with one "infected" vertex.
At each time step, we form a matching of the vertices. Then the vertices paired with infected vertices will also ...