How can i prove that any tree contains a matching of size |InternalNodes|/2? Thanks in advance
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What are internal nodes? Nodes whose degree is larger than 1? – Yuval Filmus Aug 04 '19 at 07:49
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What have you tried? Where did you get stuck? – Yuval Filmus Aug 04 '19 at 07:50
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Have you tried induction? – Yuval Filmus Aug 04 '19 at 07:50
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internal node is a node that is not a leaf. I don't event know where to start. You suggest to prove it by induction? – user108220 Aug 04 '19 at 09:27
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Hint : prove by induction that there exists a matching where all internal nodes are covered. – Tassle Aug 04 '19 at 11:41
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You can prove the following stronger claim by induction:
Given a rooted tree containing more than one vertex, there is a matching which covers all non-leaf nodes.
The proof is quite simple – we match the root to an arbitrary child, remove the edge, and recurse on the remaining rooted trees. Each remaining tree in which the root is a non-leaf in the original tree will contain more than one vertex, and so the induction hypothesis applies.
I'll let you come up with the details.
Yuval Filmus
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