26.1.2.2. Inductive Step for Sufficiency Condition

Try these first

1. 1.

   Define the term 'bipartite graph'.

   Hint

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

2. 2.

   What is a complete matching?

   Hint

   Consider how many edges are involved.

3. 3.

   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.

4. 4.

   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.

5. 5.

   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.

6. 6.

   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.