Graph Colouring Problem

Please use words to describe the solution process: Let G be a graph with exactly one cycle. Prove that x(G) is less than or equal t0 3. *(Please see attachment for proper symbols)

Propositional Logic

Verify DeMorgans laws (equation 1 and 2 below) using truth tables... Please see attached Word document.

Propositional Logic

Please see the attached Word Document.

Discrete Structures - Coloring

Let G be a properly colored graph and let us suppose that one of the colours used is red. The set of all red-coloured vertices have a special property. What is it? Graph colouring can be thought of as partitioning V(G) into subsets with this special property. (See attachment for full background)

Discrete Mathematics

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