26.2. Conclusion and References

Try these first

1. 1.

   What is Hall's Marriage Theorem?

   Hint

   Think about pairs and their connections in a graph.

2. 2.

   Define a bipartite graph.

   Hint

   Consider how edges connect different groups.

3. 3.

   What does Hall's Marriage Theorem assert?

   • A complete matching exists if |N(A)| ≥ |A| for subsets A.
   • A complete matching is impossible.
   • Every element in V1 must be connected to V2.

   Hint

   Think about matching connections.

4. 4.

   True or False: The sufficient condition is about having a complete matching.

   • True
   • False

   Hint

   Recall the theorem's implications.

5. 5.

   Provided a bipartite graph, demonstrate the failure of the matching condition. What modifications would result in a valid matching?

   Hint

   Analyze subsets closely.

6. 6.

   Construct a complex bipartite graph with 8 and 7 vertices. Illustrate using Hall's theorem whether a complete matching exists.

   Hint

   Focus on counting and verification.