Anyway, here’s an example of another proof. There’s a recipe for finding an Eulerian circuit in a graph (if such a circuit exists). It’s called Fleury’s algorithm, and there are lots of descriptions of it online. One can prove that if the degrees of the vertices are all even then the algorithm really does give an Eulerian circuit, and that (of course) implies the existence of the circuit. It’s a constructive proof. For example, you can read about this in “Graph theory” by Bondy and Murty (available online at Google books: http://books.google.co.uk/books?id=HuDFMwZOwcsC).

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