Thread: Simple graph proof question

Hi,

I have an exercise about graphs which I cannot solve. Could you please help me?

The exercise is as follows:

A simple graph, also called a strict graph, is an unweighted, undirected graph containing no graph loops or multiple edges. A well-known theorem states that the sum of the degrees of the vertices of a simple graph equals twice the number of edges of the graph.

Prove the following:

There is no simple graph with 12 vertices and 28 edges so that

(i) all vertices have degree 3 or 4
(ii) all vertices have degree 3 or 6

Thank you very much

Notice that .
So (i) is impossible.

For (ii) let s be the number of degree six vertices and t be the number of degree three.
Then .
What is wrong with that?