TDSM 10.15

There are two kinds of vertices: 1) vertices with even vertex degree; 2) vertices with odd vertex degree.

Z = sum of degrees of all vertices

X = sum of degrees of vertices with even vertex degree

Y = sum of degrees of vertices with odd vertex degree

Z = X + Y

First, Z is even. Because Z equals to 2 times the number of edges.

Second, X is even. This is obvious.

Therefore, Y is even. Because Y = Z - X. Even minus even is even.

Y even means that the sum of some odd numbers is even. But how many are the odd numbers?

(From number theory) A sum of odd numbers is only even if there is an even number of them.

Q.E.D.