Practice Case 1: K-sized Subset With More Neighbours (26.1.2.2.1)

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6 sample questions

Question 1.
What is a bipartite graph?
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Think about how the vertices are connected.
Question 2.
Define a complete matching.
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Consider what it means to cover all vertices in one set.
3Question 3.
What does Hall's Marriage Theorem state?
- A matching exists if |N(A)| = |A|
- A matching exists if |N(A)| ≥ |A| for any A
- A matching exists if all vertices are interconnected
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Consider what it means for subsets and their neighbours.
4Question 4.
True or False: If |N(A)| < |A| for any subset A, a complete matching can exist.
- True
- False
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Think about how many vertices need to be matched.
Question 5.
Prove Hall’s Marriage Theorem using a specific bipartite graph example and calculating neighbours.
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Draw the bipartite graph and label neighbours.
Question 6.
Construct different bipartite graphs, varying the connections, and identify which meet or violate Hall’s conditions.
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Compare neighbour counts with set sizes.

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