**1**

vote

**2**answers

257 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 ...

**11**

votes

**2**answers

589 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 ...

**1**

vote

**0**answers

84 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 ...

**2**

votes

**1**answer

336 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

304 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 ...

**2**

votes

**0**answers

91 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 ...

**-4**

votes

**1**answer

171 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 ...

**2**

votes

**0**answers

79 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 ...

**1**

vote

**1**answer

136 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 ...

**2**

votes

**1**answer

317 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

**2**answers

245 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 ...

**6**

votes

**1**answer

149 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 ...

**2**

votes

**1**answer

383 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

**0**answers

118 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 ...

**1**

vote

**2**answers

431 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

**0**answers

78 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 ...

**0**

votes

**1**answer

296 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 & ...

**2**

votes

**1**answer

130 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 $u$.
...

**1**

vote

**2**answers

412 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 ...

**0**

votes

**1**answer

206 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 ...

**4**

votes

**2**answers

157 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 ...

**2**

votes

**0**answers

159 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

**0**answers

94 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 ...

**8**

votes

**1**answer

347 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 $y\in Y$, ...

**4**

votes

**0**answers

149 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 $1\leq i \leq k$, ...

**6**

votes

**1**answer

826 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 ...

**2**

votes

**1**answer

231 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 ...