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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Does Ising partition function determine the number of $k$-matchings mod $4$ for cubic graphs?
Let $G$ be a cubic graph. It's known that the Tutte polynomial $T_G$ of $G$ on the hyperbola $(x-1)(y-1)=2$ determines the Ising partition function of $G$ and vice versa.
A $k$-matching in a graph $G$ ...
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How to determine if two matchings are related by a permutation?
Let $n \geq 2$ be an integer. Let
\begin{align*}
V &= \{(i, j); 1 \leq i, j \leq n \text{ and } i \neq j \} \\
E &= \{ \{v_1, v_2\}; v_1, v_2 \in V \text{ and } v_1 \neq v_2 \}.
\end{align*}
...
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Symmetric matching in special graphs
Let $G$ be a bipartite graph, $L$ ($R$) be the set of vertices in the left (right) part.
Consider a graph $T$ with the set of vertices $R \times L$ ( $L \times R$ ) in the left (right) part. For any $...
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Complexity of optimal cartesian matching
Question:
what is known about the algorithmic aspects of optimally matching a set $\mathcal{P} = \prod\limits_{i=1}^n \left(1,\,\cdots,\,k_i\right)$ of grid-points to a set of $\prod\limits_{i=1}^...
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Calculating variance-minimal perfect matchings
Question:
are there any algorithms, resp. what can be recommended, for calculating perfect matchings with the property that the variance of their edge's weights is minimal?
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Who contributed [GT13] to "Computers and Intractability"?
This is a followup to my question How does the complexity of calculating the Permanent imply the NP completeness of directed 3-cycle cover?
Question:
who contributed problem [GT13] PARTITION INTO ...
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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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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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Any updates on "The minimum cost perfect matching problem with conflict pair constraints"?
The subject of the paywalled article The minimum cost perfect matching problem with conflict pair constraints (MCPMPC) are perfect matchings of minimum cost that do not contain certain pairs of edges; ...
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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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Vertex coloring inherited from perfect matchings (motivated by quantum physics)
Added (19.01.2021): Dustin Mixon wrote a blog post about the question where he reformulated and generalized the question.
Added (25.12.2020): I made a youtube video to explain the question in detail.
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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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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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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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What is known about iterated matching as a TSP heuristic
A fairly wellknown heuristic for TSP that is based on matching is described in the 2003 paper Match twice and stitch: a new TSP tour construction heuristic by Andrew B. Kahng and Sherief Reda.
Its ...
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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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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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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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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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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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Finding minimum weight perfect matchings in sparse bipartite graphs
Question:
What can be recommended for finding optimal perfect matchings in large bipartite graphs with small vertex degree if the edge-weights are positive real values?
I am looking for ...
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Can local flip moves connect dimer matchings on 'quadrangulated' planar bipartite graphs? (perfect matching reconfiguration problem)
I'm interested in the structure of dimer matchings on planar graphs with a bipartite structure. In particular, I'm interested in whether any two perfect matchings can be connected, i.e. transformed ...
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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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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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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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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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