Sample Assignment - Mathematical Sciences Individual Project Report

1.0.ABSTRACT

The graph coloring problem is a problem in which the user is required to identify the minimum number of colors that are required to color the graph while no two same color areas share more than one point's adjacency. A significant contribution to the graph coloring is the four-color theorem. The Four Color theorem was coined by Francis Guthrie, who later shared the problem and therefore, it came to the knowledge of mathematician community. Several experts in the area tried solving the equation and proving whether they deem the theorem to be right or wrong, but most of the solvers theorem was out rightly rejected by counter proofs at some or other point of time, until the two solvers mathematicians considered using computer to solve the equation so that they could consider all prepositions and the chances of someone else later identifying a mistake be reduced. The two solvers were Appel and Haken who utilized 1200 hours on the equation to prove that the four color theorem is right and therefore any planar structure can be colored using the four colors.

Today, after so many years of the research and identification of the four-color theorem, people and companies all around the world use the theorem to solve different kinds of minimization equations. These different examples have been illustrated in the report.

2.0.INTRODUCTION AND BACKGROUND

"Two sections that share a common edge cannot be colored the same!" Nothing in the world could have turned coloring something to such a mathematical problem as this rule has and ultimately led to development of "Graph Theory" or "Graph Co louring" branch of mathematics. Graph coloring involves merely taking up coloring a graph, which could be any structure in plain or non-planar structure. Thus while graph coloring can mean coloring a map, coloring vertices or edges of a square figure, it can also mean coloring a sphere or any other 3-D figure.

The base of graph coloring is to minimize the number of colors that are required to color a specific graph. This is a simple linear programming minimization equation. And, like every other minimization equation, it too has constraint, and it is as described above, that two sections with same color shall not hold common edges. There is an exception to this constraint and that is that they can hold common edges only if it is a one-point edge. It may seem like a simple minimization problem, but it took around four generations of the mathematicians to solve it and finally accept the initial solution of Four Color Theorem.

3.0. GRAPH COLORING

Before identification of graph coloring, the significant factor worth considering in the equation is what all can be included in a graph. According to Prof. Jeremy L. Martin (2013), "A graph consists of a collection of vertices connected by edges." This means that a collection of edges and vertices is a graph, however that does means that it could be a non-planar structure too. Prof Jeremy L. Martin (2013) further describes that, "A graph is planar if its vertices and edges can be drawn as points and line segments with no crossings". And in the preposition of graph coloring that is presently accepted and formed, only planar structures are considered, since the base of graph coloring Four Color Theorem works only for planar graphs. Thus, when we take up considering coloring the portions of graph between the vertices and edges, the approach is called graph coloring. Various sets of planar and non-planar graphs have been provided in appendix 1.

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