25.1.5. Hall's Marriage Theorem

Try these first

1. 1.

   Define a bipartite graph.

   Hint

   Consider how edges connect the two sets.

2. 2.

   What is matching in a graph?

   Hint

   Think about how edges connect vertices without overlap.

3. 3.

   What is Hall's Marriage Theorem concerned with?

   • The existence of cycles in graphs
   • Conditions for complete matching in bipartite graphs
   • The total number of edges in a graph

   Hint

   Think about the relationships between vertices.

4. 4.

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

   • True
   • False

   Hint

   Consider the definitions of both terms.

5. 5.

   Given a bipartite graph with sets {A, B, C} and {X, Y, Z}, where edges connect A-X, B-Y, and C-Z, devise a complete matching. What further connections can you construct to meet Hall's condition?

   Hint

   Examine the size and distribution of edges.

6. 6.

   Consider a job assignment scenario: You have four jobs and only two employees capable of covering those roles. Why won't a complete assignment be attainable? Illustrate using Hall's theorem.

   Hint

   Count the tasks versus available workers.