Prove that no graph can have an odd number of vertices with

Prove that no graph can have an odd number of vertices with odd degrees. (The sum of an odd number of odd integers is odd.)

Solution

The sum of all degrees of any graph will be equal to twice the number of edges since each edge has one start point and one end point. Hence it gives degrees 2 to the graph.

Let the graph G have m odd number of vertices with odd degrees. Since the total number of degrees of all vertices is even, if we substract total of even degrees of vertices then the remaining will be again even.

Hence m odd vertices with odd degrees is not possible.

Hence proved.