What does the converse of Hall's Theorem state regarding bottlenecks and matches?

Conditions

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

Reasoning, step by step

1. Verify that the graph has no bottlenecks.
2. Conclude that a perfect matching exists.

Example

The current slide introduces the converse direction of Hall's Theorem for bipartite matching. It states that if a bipartite graph has no bottlenecks, then it must contain a perfect matching.

Common misconceptions

• Assuming that the absence of bottlenecks is trivial to prove as a sufficient condition for a matching.
• Thinking that the theorem provides a constructive method to find the matching.

Watch the explanation

Explore next

• What is Hall's theorem as introduced at the end of the clip?
• Why does the three-girls/two-boys example prove no matching is possible?