190Tutorial9

Tutorial 9

(Mindscapes 19 and 20, Chapter 5.3 of the textbook).

• (a) Consider the sphere on the left. It has two latitudinal circles (edges) on it and one dot (vertex) one each circle. Count the number of faces, edges and vertices on this sphere. Compute V-E+F. Did you get the answer you expected? Why, or why not? What hypothesis about the Euler Characteristic Theorem (for the sphere) does this question pertain to?

V = 2
E = 2
F = 3
2-2+3 = 3

This answer is expected because the graph is not connected. This pertains to the hypothesis that only a connected graph on a sphere follows the form of V-E+F=2.

• (b) Now add an edge that connects the two dots to the sphere of part (a) - see the sphere on the right. Now count the number of faces, edges and vertices. Compute V-E+F again. Why is this answer different from that of the calculation in part (a)?

V = 2
E = 3
F = 3
2-3+3 = 2

This answer is different because now, there is a connected graph. Connected graphs on spheres follow the equation V-E+F=2