Practice Proof of Hall's Marriage Theorem - 26.1 | 26. Proof of Hall's Marriage Theorem | Discrete Mathematics - Vol 2
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26.1 - Proof of Hall's Marriage Theorem

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Learning

Practice Questions

Test your understanding with targeted questions related to the topic.

Question 1

Easy

Define a bipartite graph.

💡 Hint: Consider how the vertices are grouped.

Question 2

Easy

What is Hall's Marriage Theorem?

💡 Hint: Think about what the neighbors of a vertex imply.

Practice 4 more questions and get performance evaluation

Interactive Quizzes

Engage in quick quizzes to reinforce what you've learned and check your comprehension.

Question 1

What condition is required for a complete matching in Hall's Theorem?

  • |N(A)| ≥ |A| for any subset A
  • |N(A)| < |A| for any subset A
  • |N(A)| = 0

💡 Hint: Recall the fundamental statement of the theorem.

Question 2

True or False: If there exists a complete matching, the neighbor condition must hold.

  • True
  • False

💡 Hint: Think about examples where this relationship applies.

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Challenge Problems

Push your limits with challenges.

Question 1

Prove that in any bipartite graph where |V1| = |V2|, if |N(A)| ≥ |A| holds for all subsets, a complete matching exists.

💡 Hint: Break down the proof into base and inductive cases, illustrating graph changes.

Question 2

Given a bipartite graph situation, create a scenario where Hall's condition fails and discuss the implications on matchings.

💡 Hint: Visualize what happens in your example and analyze neighborhood counts.

Challenge and get performance evaluation