Discrete Math : Proof that there must be 12 Pentagons on a Soccer Ball

A soccer ball is formed by stitching together pieces of material that are regular pentagons and regular hexagons. Each corner of a polygon is the meeting place for exactly three polygons. Prove that there must be exactly 12 pentagons. (Please see attachment for full question and background)

Discrete Mathematics : Prove that a 5-regular graph with 10 vertices is nonplanar.

Let G be a 5-regular graph with ten vertices. Prove that G is nonplanar.

Find values of p for which the faces of a planar p-regular graph are all triangles.

The faces of a planar p-regular graph are all triangles (that is each face has degree three). Determine, with proof, the values of p for which this is possible. (Remember a p-regular graph has all vertices of degree p).

Discrete Structures - Define and Prove

Use words to describe the solution process. No programming. 1. (a) Define a tree. (b) Define a bipartite. (c) Prove the following: Every tree is a bipartite.

Chromatic Number; Planar

Use words to describe the solution process. No programming. 2. Let G = (V,E) be a graph where V {1,2,3,4,5,6,7,8,9,10,11,12} and E contains all edges connecting to vertices a and b such that ab=0 (mod 3). What is the chromatic number of G? Is G planar?