For questions about matchings in graph theory. A matching on a graph is a set of edges such that no two edges share a common vertex.

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Induced matchings in a bipartite graph with every matching having the same number of edges

Suppose $n,k$ are positive integers such that $k\mid n$. Consider a bipartite graph $H=(A,B,E)$ such that $|A|=|B|=n$ and the edge set $E$ consists of the union of $m(H)$ induced matchings with every ...
3
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0answers
47 views

Are Bipartite Matching and General Matching Really Different Problems?

Questions: Have there been attempts to either prove or disprove, that every general matching problem can be transformed into a bipartite matching problem, from whose solution the solution of ...
2
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1answer
55 views

Problems and Algorithms Requiring Non-Bipartite Matching

While the importance of the non-bipartite matching problem itself from an algorithmic and complexity point of view is well known, applications of non-bipartite matching are hard to find. I did an ...
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0answers
43 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 $k$...
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85 views

Is there an efficient algorithm to find all the maximum matching in any tree?

A matching in a graph (G) is a set of mutually non-adjacent edges of (G). A maximum matching is a matching of maximal cardinality. A tree is an acyclic connected graph. Is there an efficient ...
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63 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 $\...
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0answers
66 views

On symmetric difference of $k$-partite perfect matchings

Suppose we have 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 ...
4
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1answer
146 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 ...
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0answers
38 views

Non-adjacent Pair of Edges with Minimal Weight Sum

Given an weighted, undirected Graph $G(V,E)$ without loops or parallel edges, what is the complexity of determining a pair of non-adjacent edges, whose sum of weights is w.l.o.g. minimal? ...
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71 views

Blossoms and Colorings

There is a striking analogy between finding maximum matchings in graphs and determining the chromatic number of graphs: both problems are fairly easy for bipartite graphs, but harder, resp. too hard ...
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14 views

Two-optimality of the Union of a Shortest Hamilton Cycle and a Minimum-weight Maximal Matching

let $G(V,E)$ be a complete, finite, symmetric and simple weighted graph with a unique shortest Hamilton cycle $T_{opt}(G)$ and a unique maximum matching of minimal weight $M_{opt}(G)$. Is it ...
3
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1answer
85 views

Probabilistic many-to-one matching

Let $p<1$ be a constant. Consider two sets $A,B$ with $n$ and $nf(n)$ vertices, respectively, where $f(n)$ is an integer. For each pair $(a,b)\in A\times B$, the edge between $a$ and $b$ appears ...
2
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2answers
143 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 ...
7
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1answer
427 views

A matching that covers vertices with maximum degree

We have a graph G with maximum degree $\Delta$. The induced subgraph on vertices with degree equal to $\Delta$ is a bipartite graph (while the original graph is not). Prove that G has a matching that ...
2
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0answers
52 views

Maximum cardinality general factor of a graph

Given a graph $G=(V,E)$ and a set of integers $B(v)$ associated to each vertex, a general factor of $G$ is a set of edges $F\subseteq E$ such that the degree of each vertex $v\in V$ in the graph $(V, ...
24
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2answers
2k views

An unfair marriage lemma

I am looking for a citeable reference to the following generalization of Hall's Marriage Theorem: Given a bipartite graph of boys and girls. In addition to gender difference, they are divided into ...
3
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0answers
135 views

A Result of Anders Bjorner: Matchings in countably infinite geometric lattices of finite height

Let $L$ be a countably infinite geometric lattice of finite height $r\ge3$. (A geometric lattice of height $r$ is an atomistic semimodular lattice such that every maximal chain has $r+1$ elements.) ...
3
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0answers
67 views

Is the size of maximum matching in vertex transitive 3-uniform hyper-graph on $n$ vertices always $\Omega(n)$?

What is the best known lower bound on the size of the maximum matching in a vertex transitive $3$-uniform hyper-graph?
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0answers
211 views

When does a “stable” assignment of buyers into goods exist?

Consider a setting of $n$ buyers and $m$ goods. We have a value matrix $V\in[0,1]^{n\times m}$ specifying how much each buyer values each good (everything is open information here and there is no ...
3
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2answers
189 views

Matching polynomials and Ramanujan graphs

Is it purely coincidental that the same number $2\sqrt{d-1}$ appears in these two following apparently disparate concepts? A $d-$regular graph is said to be called Ramanujan if its adjacency ...
3
votes
2answers
663 views

Solving assignment problem using Hungarian method vs min cost max flow problem

The traditional solution for the assignment problem is the Hungarian method - it's complexity is O(V^4) or O(V^3) if using Edmonds method. However, it can also be reduced to a min cost max flow ...
3
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1answer
491 views

Polygamous stable marriage/ assignment problem

I'm not sure under which 'algorithm' it falls under, but here is the problem: I need to match each person to 5 people from the opposite gender (each guy gets 5 girls, each girl gets 5 guys). Not all ...
5
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2answers
393 views

Matching number and chromatic number

If $G$ is a (finite) graph, denote with $\mu(G)$ the size of any maximum matching in $G$ (this number is also called the "matching number" of $G$). For odd integers $n$ we have $n=\chi(K_n) = 2\cdot\...
5
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2answers
522 views

Roots of matching polynomial of graph

At the end of this preprint, I make the following conjecture concerning the roots of the matching polynomial: If a graph $G$ is connected and contains a cycle, then the spectral radius of $G$ ...
3
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0answers
128 views

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 ...
0
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1answer
112 views

Converse of Petersen's 2-Factorization Theorem

Definition: A $k$-factor of a graph is a spanning $k$-regular subgraph. Definition: A $k$-factorization of a graph is a partition of the edge set into $k$-factors. Petersen's celebrated ...
0
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1answer
427 views

Counting matchings in a bipartite matching-covered graph

A graph is called matching-covered if every edge is containd in a perfect matching. (Such graphs are also sometimes called "elementary", e.g. in Chapter 4 of "Matching Theory" by Lovasz & Plummer)....
0
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1answer
265 views

Good lower bound on matching in bipartite graph

Suppose a bipartite graph $G=(V_1 \cup V_2, E)$ is given, and one is interested in matching vertices $V_1$ to vertices $V_2$. Assume Hall's condition does not hold, so a perfect matching does not ...
1
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1answer
88 views

Would a graph with such maximum weighted matchings exist?

Edit Tony's answer is quite nice, but I meant something else. Sorry for editing again, I meant edges. I am looking for a graph with 3 distinguished edges $xx'$,$yy'$,$zz'$ where $\deg(x)=\deg(y)=\deg(...
6
votes
1answer
406 views

How does this algorithmic proof of Edmonds-Gallai work?

Sorry, this is going to be technical and dirty. I am not looking for a proof of the Edmonds-Gallai structure theorem (I understand two of them, even if they are rather similar); I am trying to ...
3
votes
1answer
301 views

Degree conditions for k-factor

I am looking for a simple degree conditon that ensures the existence of a k-factor in a graph. The k is supposed to be relatively high and I don't mind the condition being a bit strict. Ideally, ...
2
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0answers
87 views

A non-distinct system of representative edges.

I have the following problem: Let $ \mathcal{G} = (G_{i})\_{i} $ be a collection of graphs. I would like to find a "system of representative edges" $ f : \mathcal{G} \rightarrow \bigcup_{i} E(G_{i}) $...
4
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0answers
165 views

Bounds on numbers of matchings of given sizes in bipartite graphs

I am interested in the following question: For which sets ${m_1,\ldots m_k}$ of positive integers do there exist bipartite graphs having exactly $m_i$ matchings of size $i$ for each $1\leq i \leq k$, ...
11
votes
2answers
634 views

Gale-Shapley stable marriage theorem: can we entrust matchmaking to monkeys?

Disclaimer: This is a question I have not done any real research about. I asked it myself some 5 years ago, and back then I had no idea where to start. Now I have some texts on stable matchings lying ...
6
votes
1answer
1k views

Condition on a bipartite graph to have an $m$-factor

This might be the most stupid question I am ever posting here: I am asking for a proof or a counterexample to a problem I proposed on MathLinks long ago. Let $G$ be a bipartite graph, i. e., a graph ...