All Questions
Tagged with perfect-matchings matching-theory
18 questions with no upvoted or accepted answers
10
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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 ...
- 5,409
10
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750
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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
5
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115
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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
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Counting perfect matchings with integrals
Has anyone used the Joni-Rota-Godsil integral formula (see details below) to count perfect matchings of square-grid graphs, Aztec diamond graphs, hexagon-honeycomb graphs, etc.? (Or even just to ...
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3
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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
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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
2
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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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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
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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
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35
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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; ...
- 13.2k
1
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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
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63
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$1$-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
1
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147
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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
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69
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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
1
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299
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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
0
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43
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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*}
...
- 5,215
0
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1
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171
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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 ". ...
- 163
0
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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 ...
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