Discrete Mathematics - Vol 2 | 26. Proof of Hall's Marriage Theorem by Abraham | Learn Smarter
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26. Proof of Hall's Marriage Theorem

The proof of Hall's Marriage Theorem is established, demonstrating the necessary and sufficient conditions for a complete matching exists between two subsets in a bipartite graph. The theorem states that if the number of neighbors of any subset of one partition is at least as large as the size of the subset itself, a complete matching from one partition to another is possible. Both necessary and sufficient conditions have been proven with detailed explanations along with the inductive proof strategy.

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Sections

  • 26.1

    Proof Of Hall's Marriage Theorem

    This section presents the proof of Hall's Marriage Theorem, detailing both the necessary and sufficient conditions for the existence of a complete matching in bipartite graphs.

    Learning Practice

  • 26.1.1

    Theorem Statement And Necessary Condition

    This section details Hall’s Marriage Theorem and its necessary condition for the existence of a complete matching in bipartite graphs.

    Learning Practice

  • 26.1.1.1

    Argument For Necessary Condition

    The section explores the necessary condition for the existence of complete matchings in bipartite graphs as outlined in Hall's Marriage Theorem.

    Learning Practice

  • 26.1.2

    Sufficiency Condition

    The sufficiency condition of Hall's Marriage Theorem is established through an existential proof demonstrating that a complete matching exists in a bipartite graph if the neighbor condition is satisfied.

    Learning Practice

  • 26.1.2.1

    Base Case For Inductive Proof

    The Base Case for Inductive Proof discusses Hall’s Marriage Theorem and proves both the necessary and sufficient conditions for a complete matching in a bipartite graph.

    Learning Practice

  • 26.1.2.2

    Inductive Step For Sufficiency Condition

    This section explains the inductive proof related to Hall's Marriage Theorem, focusing on the sufficiency condition for establishing the existence of a complete matching in a bipartite graph.

    Learning Practice

  • 26.1.2.2.1

    Case 1: K-Sized Subset With More Neighbours

    This section explains the necessary and sufficient conditions for complete matchings in bipartite graphs, specifically focusing on Hall's Marriage Theorem.

    Learning Practice

  • 26.1.2.2.2

    Case 2: K-Sized Subset With Exactly K Neighbours

    This section discusses Hall’s Marriage Theorem, focusing on the conditions for the existence of a complete matching in a bipartite graph.

    Learning Practice

  • 26.2

    Conclusion And References

    This section summarizes the key insights from Hall's Marriage Theorem proof and lists the references utilized in the chapter.

    Learning Practice

References

ch47.pdf

Class Notes

Memorization

What we have learnt

  • Hall's Marriage Theorem pro...
  • A complete matching exists ...
  • The proof involves both a d...

Final Test

Revision Tests