Editing Eulerian path (section)

== Directed Eulerian graphs ==
It is possible to have a [[directed graph]] that has all even out-degrees but is not Eulerian. Since an Eulerian circuit leaves a vertex the same number of times as it enters that vertex, a necessary condition for an Eulerian circuit to exist is that the in-degree and out-degree are equal at each vertex. Obviously, connectivity is also necessary. König proved that these conditions are also sufficient. That is, a directed graph is Eulerian if and only if it is connected and the in-degree and out-degree are equal at each vertex.<ref name=":0" />

In this theorem it doesn't matter whether "connected" means "weakly connected" or "strongly connected" since they are equivalent for Eulerian graphs.

Hierholzer's linear time algorithm for constructing an Eulerian tour is also applicable to directed graphs.<ref name=":0" />