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Can I create a disconnected graph with an Eulerian path? For instance does the below graph has an Eulerian path or does it have to be connected?

is this Eulerian?

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The usual definition of an Eulerian path is that it must use each edge exactly once. It does not say anything about how often vertices are visited, so yes, the cycle in your graph is an Eulerian path. (Of course you're free to work with a different concept where that all vertices must be visited, if that's what makes sense for your application). – Henning Makholm Oct 8 '11 at 16:59
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It depends on the given definition. To circumvent the problem, I tend to state the theorem as: A connected graph has an Eulerian circuit if and only if ...

You will find both definitions in books.

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