All Questions
Tagged with perfect-matchings co.combinatorics
14 questions
32
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0
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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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- 155
28
votes
3
answers
2k
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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 ...
- 82.7k
11
votes
2
answers
1k
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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 ...
- 13.2k
9
votes
1
answer
382
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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 $(n,n)$, ...
- 155
7
votes
2
answers
571
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Perfect matchings in certain classes of hypergraphs
While doing research I came unto the following problem:
Given a hypergraph $H$ which is $r$-partite, $r$-uniform (each edge contains exactly $r$ vertices), $k$-regular (each vertex is contained in ...
7
votes
2
answers
480
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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
1
answer
3k
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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
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
12
votes
2
answers
750
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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 ...
- 19.7k
8
votes
2
answers
761
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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 ...
- 13.9k
7
votes
1
answer
969
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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
7
votes
0
answers
349
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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 $\mathcal G_{2n}$.
...
- 13.9k
7
votes
1
answer
974
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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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7
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0
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203
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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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