(i) A graph G has 15 edges, 3 vertices of degree 4 and other vertices ofdegree 3. Find the number of vertices in G.

To find the number of vertices in graph G, we can use the Handshaking Theorem. The Handshaking Theorem states that the sum of the degrees of all vertices in a graph is equal to twice the number of edges.

Given that the graph has 15 edges, the sum of the degrees of all vertices is 2*15 = 30.

We know that there are 3 vertices of degree 4, so the sum of their degrees is 3*4 = 12.

Subtracting this from the total sum of degrees, we get 30 - 12 = 18. This is the sum of the degrees of the remaining vertices, each of which has degree 3.

Therefore, the number of remaining vertices is 18 / 3 = 6.

Adding the 3 vertices of degree 4, we find that graph G has a total of 3 + 6 = 9 vertices.