Practice Inductive Step for Sufficiency Condition - 26.1.2.2 | 26. Proof of Hall's Marriage Theorem | Discrete Mathematics - Vol 2
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26.1.2.2 - Inductive Step for Sufficiency Condition

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Practice Questions

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Question 1

Easy

Define the term 'bipartite graph'.

💡 Hint: Think about how the vertices are divided in Hall's theorem.

Question 2

Easy

What is a complete matching?

💡 Hint: Consider how many edges are involved.

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Interactive Quizzes

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

Question 1

What does the sufficiency condition in Hall's Marriage Theorem state?

  • A complete matching is impossible.
  • A complete matching is guaranteed if certain conditions hold.
  • The theorem applies only to even-sized graphs.

💡 Hint: Recall the details of the condition established for neighbours.

Question 2

In Case 2 of the proof, what key condition must hold regarding the size of subsets?

  • True
  • False

💡 Hint: Think about how equality plays a role in the sufficiency condition.

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Challenge Problems

Push your limits with challenges.

Question 1

In a bipartite graph with 10 vertices in V1 and 12 in V2, if each vertex in V1 connects to at least 11 vertices in V2, prove the existence of a complete matching using an appropriate case from the inductive proof.

💡 Hint: Utilize the definitions provided in Hall's theorem and parallels with your learned inductions.

Question 2

Consider a scenario where only 5 vertices in V1 have connections to V2, while others do not meet the neighbour requirements. Discuss how this affects the sufficiency condition's application.

💡 Hint: Think about how subsets interact and the importance of satisfying the connected nodes' requirements.

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