26.1.2. Sufficiency Condition

Try these first

1. 1.

   What is the necessary condition for Hall's Marriage Theorem?

   Hint

   Think about matches in pairs.

2. 2.

   Define a complete matching in the context of bipartite graphs.

   Hint

   Consider pairs—everyone must find a match.

3. 3.

   In a bipartite graph, what must be true for a complete matching to exist?

   • |N(A)| < |A|
   • |N(A)| = |A|
   • |N(A)| ≥ |A|

   Hint

   Think about matching and availability in pairs.

4. 4.

   True or False: The proof for the sufficiency condition of Hall's theorem does not use induction.

   • True
   • False

   Hint

   Remember the steps we took in class.

5. 5.

   Construct a bipartite graph with 5 vertices on one side and 3 on the other. Demonstrate whether a matching is possible based on the neighbor condition.

   Hint

   Draw edges based on connections.

6. 6.

   Using Hall's theorem, evaluate a scenario where a group of 7 students is connected to exactly 5 tutors with specific matching pairs. Discuss if a complete matching exists or not.

   Hint

   Check each student's connections before concluding.