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)

Discrete Mathematics

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

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?

discrete f3

If there is any reason why you do not want to answer the question (problem with attachment, bid price, ect.) please let me know.