All Questions
Tagged with perfect-matchings matching-theory
56 questions
0
votes
0
answers
43
views
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*}
...
- 5,215
1
vote
0
answers
35
views
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; ...
- 13.2k
3
votes
1
answer
141
views
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 ...
- 129
2
votes
0
answers
163
views
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 $(x', y') \in M$ ...
- 121
0
votes
1
answer
171
views
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 ". ...
- 163
3
votes
0
answers
232
views
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 ...
- 15.8k
1
vote
0
answers
129
views
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.
...
0
votes
0
answers
110
views
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 ...
- 2,089
1
vote
2
answers
147
views
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 $V_1\cap V_2= \emptyset$.
...
- 567
7
votes
2
answers
500
views
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 $\...
- 1,190
3
votes
1
answer
376
views
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 ...
- 181
8
votes
1
answer
384
views
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 ...
- 1,190
5
votes
1
answer
1k
views
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 ...
- 181
10
votes
0
answers
627
views
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 ...
- 5,409
3
votes
1
answer
80
views
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 \...
- 149
1
vote
1
answer
313
views
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 $G_{\pi,\pi'}$ ...
- 13.9k
3
votes
1
answer
304
views
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 ...
- 1,701
1
vote
0
answers
63
views
$1$-factorizability for linear hypergraphs with infinite edges on $\omega$
Let $H=(V,E)$ be a hypergraph. We say that $M\subseteq E$ is a matching if the members of $M$ are pairwise disjoint, and $M$ is said to be perfect if $\bigcup M = E$. Moreover, $H$ is $1$-factorizable ...
- 48.9k
3
votes
1
answer
54
views
$1$-factorizability for "complete" finite hypergraphs
Let $H=(V,E)$ be a hypergraph such that $V\neq \varnothing$ and $\varnothing \notin E$. A matching is a subset $M\subseteq E$ such that $m_1\neq m_2 \in M$ implies $m_1\cap m_2 = \varnothing$, and $M$ ...
- 48.9k
0
votes
1
answer
112
views
Weak $1$-factorizability
A simple, undirected graph is said to be $1$-factorizable if there is a partition of the edge set $E$ such that every member of the partition is a perfect matching of $G$. Let us call $G$ weakly $1$-...
- 48.9k
2
votes
1
answer
81
views
Are countable graphs with infinite minimal degree $1$-factorizable? [duplicate]
We say that a simple, undirected graph $G=(V,E)$ is $1$-factorizable if there is a partition of $E$ such that every member of the partition is a perfect matching of $G$. It is easy to see that any $1$-...
- 48.9k
5
votes
0
answers
115
views
Hypergraphs with only disjoint perfect matchings
Let $H(n,r)$ be the set of $r$-uniform hypergraph with $n$ vertices that have only disjoint perfect matchings (i.e. every hyperedge only appears in at most one of the perfect matchings). Let $m(h(n,r))...
- 155
2
votes
1
answer
321
views
Extending perfect matchings into Hamiltonian cycles
Let $G$ be a simple cubic graph which has a Hamiltonian circuit $C$. In general, it is not possible to find a second Hamiltonian circuit which contains all the chords of $C$. For example, the Wagner ...
- 1,018
0
votes
1
answer
150
views
Combining three matchings to form a maximal matching
Consider a regular tripartite graph $G$ with maximum degree $\Delta\ge3$ and parts $A,B,C$. Now, the induced subgraphs $A\cup B, B\cup C$ and $A\cup C$ are all bipartite.
Now, is there a way to ...
- 2,089
-1
votes
2
answers
306
views
Cardinality of a set of mutually disjoint perfect matchings of $K_\omega$
If $G=(V,E)$ is a simple, undirected graph, we say that $M\subseteq E$ is a perfect matching if the members of $M$ are pairwise disjoint and $\bigcup M = V$. Let $K_\omega$ be the complete graph on $\...
- 48.9k
0
votes
1
answer
358
views
A vertex transitive graph has a near perfect/ matching missing an independent set of vertices
Consider a power of cycle graph $C_n^k\,\,,\frac{n}{2}>k\ge2$, represented as a Cayley graph with generating set $\{1,2,\ldots, k,n-k,\ldots,n-1\}$ on the Group $\mathbb{Z}_n$. Supposing I remove ...
- 2,089
1
vote
0
answers
147
views
A simple case of a strong version of the Berge-Fulkerson conjecture
UPDATE 28 June 2019 A counterexample for Conjecture 2 has been provided. The conjecture is now demoted again to guess. The text has been updated to reflect this change, and there is now a new ...
- 1,018
1
vote
0
answers
69
views
Perfect matchings and edge cuts in cubic graphs - part 1
Let $G$ be a bridgeless cubic (simple) graph, and let $M$ be a perfect matching in $G$. $G-M$ will necessarily be a set of circuits. For example, if we delete a perfect matching from $K_{3,3}$ we ...
- 1,018
5
votes
1
answer
316
views
On optimal dual solutions for the minimum weight perfect matching problems in the case of metric weights
Following Lovasz-Plummer (Matching theory, North-Holland 1986, Theorem 9.2.1),
the minimum weight perfect matching problem on a complete graph
$G$ with even number of vertices and weight $w:E(G)\to
\...
- 4,878
2
votes
1
answer
338
views
Number of distinct perfect matchings/near perfect matchings in an induced subgraph
Consider a Class 1 graph with degree $\Delta\ge3$ and the induced subgraph formed by deleting a set of independent vertices of cardinality $\left\lfloor\frac{n}{\Delta}\right\rfloor$. Then, what is ...
- 2,089
10
votes
0
answers
750
views
Has this notion of vertex-coloring of graphs been studied?
In a study of a quantum physics problem, I came about an apparently very natural type of vertex colorings of a graph. The colors of the vertex $v_i$ is inherited from perfect matchings $PM$ of an edge-...
- 155
11
votes
1
answer
820
views
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 ...
- 155
-2
votes
1
answer
362
views
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 $M\subseteq E$ ...
- 48.9k
-2
votes
1
answer
74
views
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 $\delta(G) \geq n$, ...
- 48.9k
2
votes
1
answer
401
views
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?
- 48.9k
-3
votes
1
answer
125
views
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 ...
- 48.9k
7
votes
1
answer
969
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 ...
- 13.9k
8
votes
2
answers
761
views
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 ...
- 13.9k
2
votes
0
answers
64
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: ...
- 13.9k
4
votes
2
answers
987
views
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. ...
- 111
8
votes
1
answer
537
views
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 ...
- 155
5
votes
1
answer
281
views
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 ...
- 2,993
21
votes
2
answers
4k
views
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 ...
- 155
3
votes
1
answer
3k
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 ...
- 4,589
1
vote
0
answers
299
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 $k$...
- 13.9k
7
votes
2
answers
480
views
Are all numbers from $1$ to $n!$ the number of perfect matchings of some bipartite graph?
Let $f(G)$ give the number of perfect matchings of a graph $G$.
Consider set $\mathcal N_{2n}=\{0,1,2,\dots,n!-1,n!\}$.
Consider collection of all $2n$ vertex balanced bipartite graph to be $\...
- 13.9k
3
votes
0
answers
75
views
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 $\...
- 13.9k
2
votes
0
answers
365
views
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 ...
- 13.9k
4
votes
1
answer
592
views
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 ...
- 239
2
votes
2
answers
354
views
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 ...
- 239