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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).2011-10-08

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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.