**1**

vote

**0**answers

67 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 ...

**3**

votes

**0**answers

59 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 ...

**0**

votes

**0**answers

57 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**

votes

**1**answer

140 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 ...

**0**

votes

**0**answers

31 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?
...

**2**

votes

**0**answers

58 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 ...

**0**

votes

**0**answers

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**

votes

**1**answer

81 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**

votes

**2**answers

135 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**

votes

**1**answer

371 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**

votes

**0**answers

51 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, ...

**23**

votes

**2**answers

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**

votes

**0**answers

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.)
...

**2**

votes

**0**answers

63 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?

**5**

votes

**0**answers

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**

votes

**2**answers

181 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

**2**answers

618 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**

votes

**1**answer

468 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**

votes

**2**answers

381 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) = ...

**5**

votes

**2**answers

505 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**

votes

**0**answers

124 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**

votes

**1**answer

110 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**

votes

**1**answer

410 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 & ...

**0**

votes

**1**answer

258 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**

vote

**1**answer

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 ...

**6**

votes

**1**answer

403 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

**1**answer

300 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**

votes

**0**answers

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**

votes

**0**answers

164 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

**2**answers

630 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

**1**answer

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 ...