Why does the three-girls/two-boys example prove no matching is possible?

Conditions

• A bipartite graph with left set G and right set B.
• A highlighted subset S of three girls.
• Their collective neighborhood $E(S)$ contains only two boys.

Reasoning, step by step

1. Identify the subset S of three girls.
2. Determine the neighborhood $E(S)$ of these three girls.
3. Compare the sizes: |S| = 3 and |$E(S)$| = 2.
4. Apply the Bottleneck Lemma: since |S| > |$E(S)$|, no match is possible.

Example

The diagram shows that those three girls connect only to two boys on the right. The speaker explains that two boys would have to be shared among three girls, so there are not enough compatible boys for all three girls to be matched distinctly.

Common misconceptions

• Thinking that a matching fails only if the whole right side is too small.
• Assuming that the other two girls on the left can somehow compensate for the lack of boys for this specific subset.

Watch the explanation

Explore next

• How is a bottleneck defined in this video?
• What does the Bottleneck Lemma say?