Proof
(related to Corollary: Even Number of Vertices with an Odd Degree in Finite Graphs)
- Let \(G=(V,E,\gamma)\) be a finite graph and let \(O\subseteq V\) be the subset of all vertices with odd degree.
- Then \(V\setminus O\) is the subset of all vertices with even degree.
- According to the handshaking lemma we obtain \(\sum_{v\in V}\deg_G(v)=2|E|.\)
- It follows \[\sum_{v\in O}\deg_G(v)=2|E|-\sum_{v\in V\setminus O}\deg_G(v).\]
- Because the right side of the equation is even, so must be the left side.
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References
Bibliography
- Krumke S. O., Noltemeier H.: "Graphentheoretische Konzepte und Algorithmen", Teubner, 2005, 1st Edition
