An Eulerian orientation of a finite graph that is undirected is a graph orientation in which every graph vertex has equal indegree and outdegree (Finch 2003, p. 412). Multiple edges are oriented individually. A graph loop contributes one to both the indegree and outdegree and contributes two to the vertex degree.
A finite graph that is undirected admits an Eulerian orientation iff every graph vertex has even vertex degree. Necessity follows because the vertex degree is the sum of the equal indegree and outdegree. For sufficiency, orient an Eulerian cycle in each connected component that contains a graph edge. Isolated vertices require no directions. An Eulerian orientation of a connected graph with at least one graph edge is an Eulerian digraph.
The cycle graph 
, for 
, has two Eulerian orientations, one in each direction
around the graph cycle. A graph
with no edges has one Eulerian orientation.
