Define a geodesic triangulation of an abstract closed smooth surface. Define the Euler number of a triangulation, and state the Gauss-Bonnet theorem for closed smooth surfaces. Given a vertex in a triangulation, its valency is defined to be the number of edges incident at that vertex.

(a) Given a triangulation of the torus, show that the average valency of a vertex of the triangulation is 6 .

(b) Consider a triangulation of the sphere.

(i) Show that the average valency of a vertex is strictly less than 6 .

(ii) A triangulation can be subdivided by replacing one triangle $\Delta$ with three sub-triangles, each one with vertices two of the original ones, and a fixed interior point of $\Delta$.

Using this, or otherwise, show that there exist triangulations of the sphere with average vertex valency arbitrarily close to 6 .

(c) Suppose $S$ is a closed abstract smooth surface of everywhere negative curvature. Show that the average vertex valency of a triangulation of $S$ is bounded above and below.