All Questions
Tagged with perfect-matchings co.combinatorics
23 questions with no upvoted or accepted answers
32
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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
8
votes
0
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245
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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 ...
- 13.9k
7
votes
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 ...
- 965
7
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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 $\mathcal G_{2n}$.
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- 13.9k
6
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0
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375
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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 ...
- 1,343
6
votes
0
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154
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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:
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- 25.4k
4
votes
0
answers
187
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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!\...
- 13.9k
3
votes
0
answers
232
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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 ...
- 15.8k
3
votes
0
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75
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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 $\...
- 13.9k
3
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0
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88
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Perfect Matching for Edge-transitive Hypergraphs
I'm new to this subject, but I've noticed that a lot of work has been done on perfect matching in k-uniform hypergraphs. I'm curious to know if there are any results on perfect matching in the more ...
- 31
2
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0
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124
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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 $...
2
votes
0
answers
163
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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 $(x', y') \in M$ ...
- 121
2
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0
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108
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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 ...
- 13.9k
2
votes
0
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60
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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 ...
- 13.9k
2
votes
0
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64
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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
2
votes
0
answers
365
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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 ...
- 13.9k
1
vote
0
answers
129
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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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1
vote
0
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94
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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 ...
- 13.9k
1
vote
0
answers
53
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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 $n$ ...
- 13.9k
1
vote
0
answers
66
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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?
- 13.9k
0
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0
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84
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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 ...
- 3,499
0
votes
0
answers
110
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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 ...
- 2,089
0
votes
0
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54
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Do all induced subgraphs of powers of cycles have a perfect matching
Do all independence induced subgraphs of powers of cycles have a distinct 1-factor? By independence induced, I mean those induced subgraphs which are formed by removing a maximal independent set of ...
- 2,089