It is also possible to solve hipartite matohing problens aith the ford$-$

Fulkerson algorithe. Sugpose ve are gizen a bipartite graph as belox.

In order to find a bipartite natching, we can turn this into a network

Flow problon. Add a source node s: and a sink node al ahere sa is

Gomected to each verter in the first partition and si is oonnected to

Every vertea in the scoond partition ksign to each edge a direction.

As in the folloaing zraph, and also assign to each edge a oapacity of

$1$ and a flow of $0.$ By applying the Ford$-$Falkerson al gorithe, finding

A naxirun flon will also give us a naxirun matohing. consisting of

All edges between the two parts of the graph that are assigred a flow

of $1.$

Now monder the followine Hipertite grape:

$($a$)$ Use the bipartite matching algorithe to find a ratiman natching

of the graph.

$($b$)$ Use Fard$-$Falkerson to find a ranimer natching. At each step.

Irficate clearly hon you have labelied the vertices