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A Graph containing an Eulerian Circuit. An undirected Graph is Eulerian Iff every Vertex has Even Degree. A Directed Graph is
Eulerian Iff every Vertex has equal Indegree and Outdegree. A planar
Bipartite Graph is Dual to a planar Euler graph and vice versa. The number of Euler graphs with
nodes are 1, 1, 2, 3, 7, 16, 54, 243, ... (Sloane's A002854).
References
Sloane, N. J. A. Sequence
A002854/M0846
in ``An On-Line Version of the Encyclopedia of Integer Sequences.''
http://www.research.att.com/~njas/sequences/eisonline.html and Sloane, N. J. A. and Plouffe, S.
The Encyclopedia of Integer Sequences. San Diego: Academic Press, 1995.