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?