Some theorems on the Euler's path and the Euler's circuit, inference

Some definitions of the Euler pathway and the Euler circuit:

No-show diagram:
G is a connected undirected graph.
(1) Once per side of the G and only once the path is Oraton (the starting and ending points are not necessarily the same).
(2) If the Oraton road is a loop (the starting and ending points are the same), then the Euler circuit.
(3) The non-direction graph G with the Euler circuit is called Eulerian graph.

To map:
D is a direction graph, D Jhoira (to change the forward side of D to no edge) is connected
(1) A path that passes through each edge of D once and only once is called the Oraton Road (the starting and ending points are not necessarily the same).
(2) If there is a return to the Oraton road (the beginning and end of the same), then called the Oraton Road.
(3) A direction diagram D with a forward Euler pathway is called a Eulerian graph.

For the determination of the Euler's path and the Euler's circuit:

undirected graph G is a sufficient and necessary condition for the existence of Euler pathways: G is a connected graph, and G has only two singular nodes (those with an odd degree of degrees) or no singularities.
Inference 1: When the undirected graph G is a connected graph with two singularities, the Euler path of G must be the end point of the two nodes.
Corollary 2: When undirected graph G is a connected graph without singularities, G must have a Euler loop.
Corollary 3: There is a sufficient and necessary condition for the existence of a Euler loop for undirected graph G: G is a connected graph with no singularities, and G has only two singular nodes (those with an odd degree of degrees) or no singularity nodes.

There are sufficient and necessary conditions for the directed Euler pathway in directed graph D: D is a directed graph, the base diagram of D is connected, and all vertices are equal in degrees (case 1), or, in addition to two points, the degrees of the remaining vertices are equal to the degrees of entry, and in both vertices, the difference between the degree of the one vertex and the Another out-of-order difference is 1 (conditions 2).
Inference (1): Condition 1 shows that there is a direction to the Euler loop.
Inference (2): The case 2 shows that there is a direction to the Oraton road, the path to the difference between the degree and the degree of 1 of the vertex as the starting point, with the difference between the degree and the degree of 1 vertex as the end point.
Inference (3): There is a sufficient and necessary condition for the existence of a forward Euler loop for the direction diagram D: The base diagram of D is a connected graph, and the access and ingress of all vertices are equal.

Some theorems on the Euler's path and the Euler's circuit, inference