It is proved that if G(V, e) is a directed strongly connected graph, the following properties are equivalent:

(i)G has an Euler path, that is, a closed trace containing all the edges in G..

(ii) The in degree of each vertex in V is equal to the out degree.

(iii) The edge set of G can be divided into cycles.

A strongly connected graph refers to a graph in which there is a path from v 1 to v2 and a path from v2 to v 1 between any two points in a directed graph.

Closed circuit: A closed circuit with different sides. (Closed loop: a path whose starting point and ending point are at the same point)

In-degree: A point in a directed graph is the sum of the times of the end point of an edge in the graph.

Range: the sum of the times that a point in a directed graph serves as the starting point of an edge in the graph.

A cycle refers to a path with different points and edges except that the starting point is equal to the ending point.

A closed trace whose vertices are not repeated is called a cycle.