Practice - Inductive Step for Sufficiency Condition
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Practice Questions
Test your understanding with targeted questions
Define the term 'bipartite graph'.
💡 Hint: Think about how the vertices are divided in Hall's theorem.
What is a complete matching?
💡 Hint: Consider how many edges are involved.
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Interactive Quizzes
Quick quizzes to reinforce your learning
What does the sufficiency condition in Hall's Marriage Theorem state?
💡 Hint: Recall the details of the condition established for neighbours.
In Case 2 of the proof, what key condition must hold regarding the size of subsets?
💡 Hint: Think about how equality plays a role in the sufficiency condition.
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Challenge Problems
Push your limits with advanced challenges
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.
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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