25.2. Conclusion and Summary

Try these first

1. 1.

   Define a bipartite graph.

   Hint

   Focus on how the vertices are organized.

2. 2.

   What is a matching?

   Hint

   Think about how edges can connect without overlapping.

3. 3.

   What is the key requirement for a complete matching according to Hall's Marriage Theorem?

   • Each vertex in the first set must be matched to a unique vertex in the second set.
   • The neighbors count must be at least equal to the number of vertices in any chosen subset.
   • All edges must create cycles.

   Hint

   Think about how many connections you need for job assignments.

4. 4.

   True or False: Every maximum matching is also a maximal matching.

   • True
   • False

   Hint

   Consider the definitions of both terms.

5. 5.

   Given a bipartite graph with five jobs and only three employees, analyze whether a complete matching exists using Hall’s theorem.

   Hint

   Check the neighbors for each subset of jobs.

6. 6.

   Create your own example of a bipartite graph that demonstrates a maximum matching different from a complete matching.

   Hint

   Include at least one unmatched vertex in your example.