The term "Euler graph" is sometimes used to denote a graph for which all vertices are of even degree (e.g., Seshu and Reed 1961). Note that this definition is different from that of an Eulerian graph, though the two are sometimes used interchangeably and are the same for connected graphs.
The numbers of Euler graphs with 
, 2, ... vertices are 1,
1, 2, 3, 7, 16, 54, 243, 2038, 33120, ... (OEIS A002854;
Robinson 1969, Mallows and Sloane 1975; Buekenhout 1995, p. 881; Colbourn and
Dinitz 1996, p. 687), the first few of which are illustrated above. There is
an explicit formula giving these numbers.
There are more Euler graphs than Eulerian graphs since there exist disconnected graphs having
multiple disjoint cycles with each graph
vertex of even degree
but for which no single graph cycle passes through
all edges. The numbers of Euler-but-not-Eulerian graphs
on 
, 2, ... vertices
are 0, 0, 0, 0, 0, 1, 2, 7, 20, 76, 334, 2498, ... (OEIS A189771),
the first few of which are illustrated above.
