All Questions
9 questions
2
votes
0
answers
163
views
Generalizing Hall's marriage theorem to non-perfect matchings
Let $G = (X, Y, E)$ be bipartite graph such that $|X|=|Y|=n$.
A matching $M \subseteq E$ is a subset of disjoint edges
(i.e., there does not exist a pair of edges $(x, y) \in M$ and $(x', y') \in M$ ...
- 121
5
votes
1
answer
1k
views
Bipartite graph with exactly one perfect matching
$\textbf{Problem:}$ Find all bipartite graphs $G[X,Y]$ satisfying the following properties:
$1.$ $|X|=|Y|$, where $|X|\ge 2$ and $|Y|\ge 2$.
$2.$ All vertices have degree three except for two vertices ...
- 181
1
vote
1
answer
313
views
Unique bipartite perfect matchings and cycles?
Given a graph $G$ which is bipartite and balanced and has unique perfect matching let $G^{e}$ be $G$ without edge $e$. Let $G\cup G_{\pi,\pi'}$ be union of $G$ and $G_{\pi,\pi'}$ where $G_{\pi,\pi'}$ ...
- 13.9k
7
votes
1
answer
969
views
Graph to Bipartite conversion preserving number of perfect matchings
Given a graph $G$ on $n$ vertices is there a technique to convert to a balanced bipartite graph $B$ with $O(n^c)$ vertices at some fixed $0<c$ in $O(n^{c'})$ time at some fixed $0<c'$ such that ...
- 13.9k
4
votes
2
answers
987
views
Applications of Hafnians
I am learning about Hafnians but I am struggling to find real-world applications of them. I understand the applications of determinants, permanents, and even pfaffians but I am at a loss for Hafnians. ...
- 111
3
votes
0
answers
75
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 $\...
- 13.9k
2
votes
0
answers
365
views
On symmetric difference of $k$-partite perfect matchings
Given 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 symmetric ...
- 13.9k
0
votes
1
answer
773
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)....
- 6,962
8
votes
1
answer
2k
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 ...
- 33.8k