**Combinatorial and Discrete Geometry: Sheet M (Matchings)**

Prepared by:

Joseph Malkevitch

Department of Mathematics and Computer Studies

York College (CUNY)

Jamaica, New York 11451

email: joeyc@cunyvm.cuny.edu

web page: http://www.york.cuny.edu/~malk

Definitions

1. If G is a graph a *matching* M is a collection of disjoint edges of G. (Two edges are disjoint if they have no vertex in common.)

2. If M is a matching of G, M is said to be a *maximum cardinality matching* if no matching of G has more edges than M.

3. If M is a matching of G, M is said to be a *maximal matching* if no edge of G can be added to M and still be a matching.

4. A matching M of G is said to be *perfect* if every vertex of G is a vertex of some edge in the matching.

For each graph below:

a. If possible find a matching with 4 edges.

b. If possible find a maximal matching with 3 edges.

c. If possible find a perfect matching.

d. Find a maximum cardinality m

G:

H:

I:

J:

K:

L: