What is Hall's theorem as introduced at the end of the clip?

Conditions

• Applies to bipartite graphs.
• Requires the condition of having no bottlenecks.

Reasoning, step by step

1. Check if the graph has any bottlenecks.
2. If there are no bottlenecks, conclude that a matching exists.

Example

The slide title reads Hall's Theorem and the statement reads Conversely, if there are no bottlenecks, then there is a match. Not obvious. Discuss next.

Common misconceptions

• Assuming that the converse of the bottleneck lemma is equally obvious to prove.
• Thinking that Hall's theorem provides the algorithm to find the matching.

Watch the explanation

Explore next

• Why does the three-girls/two-boys example prove no matching is possible?
• What does the converse of Hall's Theorem state regarding bottlenecks and matches?