The necessary and sufficient conditions for the existence of Euler circuits and Euler pathways and their proofs

Theorem 1: The necessary and sufficient condition for the presence of a Euler loop in a connected graph is an even number of vertices in the connected graph.

First, we prove adequacy, that is, the existence of Euler loops, the degree of all vertices in the graph must be an even number. Take a point in the diagram, take that point as the starting point, follow the Euler loop, the current vertex is out of 1, and then through the other vertices, notice that if the Euler path passes through a vertex (including the starting point), it must leave this point, so that the sum of the degrees of access is even, until all the edges are traversed, the end of Make the starting point into the degree plus 1, so after the beginning of the degree and become even, the Euler Loop end (note that we have not explained the assumption that the number of edges is poor, so this process must end).

Secondly, we have to prove the necessity, that is, if all the vertices in the connected graph have an even number of degrees, there must be a Euler loop. We illustrate this by means of a constructive proof of existence (here are some descriptions of the methods of proof). First, we find a circuit in the connected graph (the selection of the loop is arbitrary and can always be found, the proof of the adequacy of the above can effectively illustrate this point), if this circuit is Euler circuit, then the conclusion has been established, otherwise, we delete all the edges in the loop, the isolated vertex will ignore it, Then the sub-graph (not necessarily connected, and still satisfies all the vertices of the degree is an even number of properties) and the deleted loop must have a common vertex (the graph of connectivity to ensure this), with the point as a starting point to continue to find the loop, and then delete, continuation of this method, Until all the edges have been removed (as in the proof of adequacy, the number of edges ensures that the process is bound to end), all of these deleted loops are connected together to form a Euler loop.

At this point, we have completed the existence of the Euler circuit proof of the necessary and sufficient conditions, and it should be noted that in the structural existence of proof we give a search for the Euler loop algorithm process.

Next, we prove the following theorem.

Theorem 2: The necessary and sufficient condition for the existence of Euler pathways in a connected graph is an odd number of degrees with only two vertices in the connected graph.

Still first to prove adequacy, that is, there are Ouratonlo in the figure and only two vertices of the degrees are odd, the other vertices are an even number of degrees, noting that because the starting point and the end point is different, so the beginning and end of the Euler path is bound to be two odd-numbered vertices, in addition, there is no other odd-numbered vertices, As we walk along the starting point of the Euler pathway, as long as a vertex is bound to leave the vertex, an in-line edge is paired with an out-of-the-way edge, contributing even degrees to the vertex until it reaches the end point (or, of course, it may leave again, as long as the end has not been traversed).

Next, we prove the necessity, that is, there are only two odd vertices in the connected graph, there must be Oraton road, how to prove this point? A very ingenious way is to make Euler's path into Euler circuit, in other words, we connect two odd vertices, so that all the vertices in the connected graph is an even number of degrees, by the theorem 1 just proved that the connectivity diagram exists in the Euler loop, notice that only our own increase of the auxiliary edge deleted, proves the existence of Euler's pathway, and once again we prove it with the proof of the existence of tectonics.

Finally, a stroke of intelligence problem is weak burst feeling has wood:).

Above.

The necessary and sufficient conditions for the existence of Euler circuits and Euler pathways and their proofs