**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 ...

**8**

votes

**1**answer

351 views

### When does a `distinguished matching' exist?

Suppose I have a bipartite graph on a pair of vertex sets $X$ and $Y$.
Definition: A distinguished matching is a subset $DX\subseteq X$ and a subset $DY\subseteq Y$ such that:
For all $y\in Y$, ...

**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 ...

**6**

votes

**2**answers

270 views

### extremal bipartite graph

I'm facing the following question:
Given a bipartite graph $G = (L \cup R, E)$.
Let $n = |L|$, $m = |R|$, and a parameter $k \in \mathbb{N}$, $n > m > k$.
What is a minimal possible number ...

**6**

votes

**1**answer

120 views

### Analysis of the Laplacian of a random bipartite graph

My analysis of an engineering problem reduced to analysis of the Laplacian of a (random) bipartite graph. There are a few particular questions I am interested in, but not sure which direction to take ...

**6**

votes

**1**answer

157 views

### Does high min degree and high odd girth imply near bipartiteness?

Say $G$ has odd girth at least $k$ and min degree $2n/k$. There is a classical result by Andrasfai, Erdos, and Sos that says that $G$ is bipartite. (Odd girth is the length of the shortest odd cycle ...

**5**

votes

**2**answers

196 views

### Bipartiteness criterion

A graph is bipartite if and only if it does not contain odd cycles. Is there a similar criterion for hypergraphs? (A hypergraph is called bipartite if its vertices can be colored in two colors so that ...

**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 $1\leq i \leq k$, ...

**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 ...

**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

**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 ...

**2**

votes

**1**answer

100 views

### A graph assignment problem

Consider bipartite graph with vertex set $V_1\cup V_2$ where $|V_1|=\frac{n(n-1)}2$ and $|V_2|=n$. The vertices in $V_1$ all have degree $2$ and connected to two vertices in $V_2$. The vertices in ...

**2**

votes

**1**answer

533 views

### bipartite graph coloring

Hi, I don't know much about graph theory so I would need to know if the following problem has a positive answer or a reference. There is a bipartite graph G with the two vertex sets V1, V2. Each ...

**2**

votes

**1**answer

236 views

### Bipartite graphs with prescribed Matching $M$ and genus $g$.

Let $B_{n,n}$ be a bipartite graph on $2n$ vertices with $n$ vertices of each color.
Given two integers $g$ and $M$, construct the smallest genus $g$ $B_{n,n}$ with exactly $M$ matchings.
My first ...

**2**

votes

**1**answer

331 views

### Obstructions to genus $g+1$ bipartite graph having genus $g$

Say $B_{n,n}$ is a bipartite graph on $2n$ vertices with each color assigned to $n$ vertices.
Say I know $g \le \operatorname{genus}(B_{n,n}) \le g+1$. What obstructions prevent $B_{n,n}$ from being ...

**2**

votes

**1**answer

107 views

### Bipartite dimension of an almost crown graph

A crown graph is a complete bipartite graph from which a perfect matching has been removed.
The bipartite dimension of a graph is the minimum number of complete bipartite subgraphs needed to cover ...

**2**

votes

**1**answer

133 views

### Searching for equal subsets in a bipartite graph

Let $(U,V)$ be a finite bipartite graph of two parts $U$ and $V$. For any subset $u\subset U$ define the image $Im(u)$ in $V$ consisting of all vertices of $V$ connected to at least one vertex of $u$.
...

**2**

votes

**2**answers

352 views

### Finding maximal k-degenerate subgraphs

Given a graph $G$, let $H$ be a $k$-degenerate (not necessarily induced) subgraph of maximal size. Are there any known lower bounds on $|E(H)|$ for particular classes of $G$ and values of $k$?
I've ...

**2**

votes

**0**answers

36 views

### What is the densest bipartite graph with unique Hamiltonian cycle?

In a prior post regarding perfect matching, it was stated that the densest graph with a perfect matching cannot have more than $n^2$ edges, if graph has $2n$ vertices.
Analogously, what is the ...

**2**

votes

**0**answers

71 views

### Maximum number of $4$-cycles

Suppose we have a balanced bipartite planar maximum degree $k$ graph.
How many such graphs on $2n$ vertices have at most $f(n)$ maximum number of $4$ cycles for a given function $f:\Bbb ...

**2**

votes

**0**answers

76 views

### Regularity for a bipartite graph

Let $G$ be a bipartite graph with $2^n$ left vertices and $2^n$ right vertices such that:
1) degree of every vertex is not greater then $2^t$
2) number of all edges is greater than $2^{n +t - O(\log ...

**2**

votes

**0**answers

131 views

### Minimum number of perfect matchings in a regular bipartite graph

Is there a lower bound on the number of perfect matchings in a $k$-regular bipartite graph?
One can use Hall's marriage theorem and induction on $k$ to derive the lower bound of $k$. I can't come up ...

**2**

votes

**0**answers

90 views

### Characterize the equivalence class of bipartite graphs obtained from each other by elementary row operations on their adjacency matrices

Let $M$ be an $m\times n$ matrix real matrix. Let $G$ be a bipartite graph, with partitions $A$ and $B$, such that $|A|=m$ and $|B|=n$. A node $i\in A$ is linked to a node $j\in B$ if and only if ...

**2**

votes

**0**answers

126 views

### Bipartite independence number

Consider a balanced bipartite graph $G=(U,V,E)$, i.e., a bipartite graph with $|U|=|V|$. An independent set $I$ of $G$ is balanced if $|I \cap U| = |I \cap V|$.
The bipartite independence number of ...

**2**

votes

**0**answers

178 views

### Structure of almost all bipartite graphs

I am studying some properties related to bipartite graphs and it would be useful for me to know if there is anything known about the structure of almost all bipartite graphs. For example, is it true ...

**1**

vote

**2**answers

509 views

### Enumeration of labeled connected bipartite graphs given partite sets

What would be the closed-form expression defining number of all possible labelled connected bipartite graphs given $\mid X \mid = m, \mid Y \mid = n - m $?

**1**

vote

**3**answers

110 views

### On number of perfect matchings

Consider $2n$ vertex balanced bipartite graph.
If total number of edges is $n^2$ then we have $n!$ perfect matchings.
Fix $c\in(0,\frac12)$ and consider collection of $2n$ vertex balanced bipartite ...

**1**

vote

**2**answers

353 views

### Expected matching in a bipartite graph

Consider a random bipartite graph constructed on vertex classes of size $n$ with each edge present independently with probability $p$. How could I go about calculating the size of the expected ...

**1**

vote

**1**answer

145 views

### Sequences that represent different drawing of chords?

In combinatorics there are there are special kind of sequences, in which their terms represent the number of different ways that we can draw chords with some properties.
Actually my question is ...

**1**

vote

**2**answers

484 views

### Similarity measure between 2 bi-partite graph.

Hello there, i need to solve this problem:
I have 2 different bi-partite weighted graph, g1 and g2 and i would like to measure their similarity, g1 and g2 may have different number of vertex and edges ...

**1**

vote

**1**answer

54 views

### Counting bounded genus non-isomorphic graphs

What is the number of non-isomorphic $2n$ vertex balanced bipartite graphs of degree at most $d$ and genus $g$?
I am most interested in $d\leq3$ and $g=0$.

**1**

vote

**0**answers

23 views

### Expected number of perfect matchings in bounded degree bipartite graphs

Consider collection $\mathcal C_{n,n,\Delta}$ of every $2n$ vertex balanced bipartite graph of average degree $\Delta$.
What is the expected number of perfect matching a graph in $\mathcal ...

**1**

vote

**0**answers

91 views

### Curve associated to bipartite graph

Given real biadjacency matrix $A\in\{0,1\}^{n\times n}$ of a bipartite graph with rank $r\in[2,n-1]$, denote $A(x)$ to be matrix where $0$ is replaced by $x$ and $1$ by $1-x$. Denote ...

**1**

vote

**0**answers

92 views

### Details about Kelman's equivalent form of Barnette's conjecture

Barnette's conjecture states that every cubic planar bipartite 3-connected graph admits Hamiltonian cycles.
Kelman claims that this conjecture is equivalent to a stronger one, which imposes some ...

**1**

vote

**0**answers

103 views

### How many extreme maximal cliques are in an n*m 0-1 matrix?

We can use an n*m 0-1 matrix to denote a bipartite graph. Mining maximal bicliques in such matrix is an open problem.
The extreme maximal clique is a special maximal clique. A clique in such matrix ...

**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 ...

**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

**0**answers

41 views

### 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?

**0**

votes

**0**answers

73 views

### Perfect matchings that never combine to form cycle union cover

Can there be exponential number (in terms of vertices) of perfect matchings on a bipartite graph so that no combination of them yields a union of cycles that cover all vertices?
Is there a way to ...

**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

183 views

### Bipartite graph [closed]

First of all, thank you for your time to reading my post.
I am a researcher but not a mathematician, i have some difficulties in solving a math problem, that why i am here to ask your help. I just ...