Combinatorial and Discrete Geometry: Sheet M (Matchings)

Prepared by:

Joseph Malkevitch
Department of Mathematics and Computer Studies
York College (CUNY)
Jamaica, New York 11451

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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