# Tagged Questions

**6**

votes

**1**answer

142 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 ...

**1**

vote

**0**answers

95 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

325 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 $?

**0**

votes

**1**answer

162 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 & ...

**4**

votes

**2**answers

151 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

122 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 ...

**8**

votes

**1**answer

340 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 $y\in Y$, ...

**2**

votes

**2**answers

148 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 ...

**9**

votes

**2**answers

465 views

### Gale-Ryser 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 ...