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Does the graph with only one vertex has an Eulerian path?
And i think he has an Hamiltonian path, right?

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I don't know if everyone defines it the same way, but Diestel's book, at least, allows a path to consist of a single vertex, which means that the answer is yes. –  Andrew Uzzell Dec 12 '12 at 14:48
@Andrew: You could post that as an answer so the question doesn't remain unanswered. –  joriki Dec 12 '12 at 17:34
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I'm not sure if all graph theory books treat degenerate cases the same way, but Diestel's Graph Theory, at least, allows a path to have length $0$, i.e., to consist of a single vertex with no edges. If a graph consists of a single vertex $v$, then the a path consisting of $v$ is vacuously Eulerian. It is also a Hamiltonian path, since it contains all of the vertices of the graph.

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