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Wikipedia's Eulerian Path states,

An undirected graph has an Eulerian cycle if and only if every vertex has even degree, and all of its vertices with nonzero degree belong to a single connected component.

But I find a mistake with this. The beginning, S, and end, T, vertices could be joined by single edges to the graph, G, where every vertex (beside S & T) has even degree. You would still be able to visit every edge once, like:

Start at S, go to a vertex in G, Ga, do a Euler tour in G finishing on vertex Gb, end by going from Gb to T.

S and T only have degree 1.

Or have I made a mistake?

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    $\begingroup$ The definition of an Eulerian cycle is that it ends at the same point that it begins at. Apart from this, your reasoning is correct. $\endgroup$
    – Mathmo123
    Commented Jun 22, 2014 at 12:45

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You have found why a graph might have an Eulerian path but no Eulerian cycle.

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