A simple graph has no edges connected from a node to itself, so the total of the degrees of all the vertices involved will be 2*18 = 36. The total of the degrees of the degree 3 vertices is 12, so the total of the degrees of the degree 4 vertices must be 36 - 12 = 24. There must be 24/4 = 6 vertices of degree 4, making a total of

You can construct such a graph as follows. Identify the vertices with numbers 1 through 10. Make the following connections:

(1-2-3-4-5-6-7-8-9-10-1)

(2-4)

(7-9)

(1-5-8-1)

(3-6-10-3)

Vertices 2, 4, 7, 9 will have degree 3. The remainder have degree 4. (There are many other possibilities.)

**10 vertices**in the graph.You can construct such a graph as follows. Identify the vertices with numbers 1 through 10. Make the following connections:

(1-2-3-4-5-6-7-8-9-10-1)

(2-4)

(7-9)

(1-5-8-1)

(3-6-10-3)

Vertices 2, 4, 7, 9 will have degree 3. The remainder have degree 4. (There are many other possibilities.)