Graph Theory and Graph Coloring Lindsay Mullen (Abstract) Algebra and Number Theory Combinatorics (Discrete Mathematics) Graph Theory Graph Coloring

What is Graph Theory? Branch of mathematics concerned with networks of points connected by lines The subject of graph theory had its beginnings in recreational math problems, but it has grown into a significant area of mathematical research with applications in chemistry, operations research, social sciences, and computer science. The history of graph theory may be specifically traced to 1735, when the Swiss mathematician Leonhard Euler solved the Königsberg bridge problem.

Königsberg Bridge Problem A puzzle concerning the possibility of finding a path over every one of seven bridges that span a forked river flowing past an island—but without crossing any bridge twice.

In 1735 the Swiss mathematician Leonhard Euler presented a solution to this problem, concluding that such a walk was impossible. It would be nearly 150 years before mathematicians would picture the Königsberg bridge problem as a graph consisting of nodes (vertices) representing the landmasses and arcs (edges) representing the bridges. The degree of a vertex of a graph specifies the number of edges incident to it. In modern graph theory, an Eulerian path traverses each edge of a graph once and only once. Thus, Euler’s assertion that a graph possessing such a path has at most two vertices of odd degree was the first theorem in graph theory. Led to the development of the branches of mathematics known as topology and graph theory

Graphs In elementary mathematics, "graph" refers to a function graph or "graph of a function," i.e., a plot. As used in graph theory, the term graph does not refer to data charts such as line graphs or bar graphs. Instead, it refers to a set of vertices (that is, points or nodes) and of edges (or lines) that connect the vertices. Graph - a visual representation of edges and vertices Edge - the line between two boundaries Vertex - a point where two or more lines meet

Types of Graphs Multigraph – multiple edges are allowed between vertices Simple graph - a graph without loops and with at most one edge between any two vertices Unless stated otherwise, graph is assumed to refer to a simple graph. Pseudograph - A graph that may contain multiple edges and graph loops

Simple graph in terms of edges and vertices

Graph Coloring The assignment of labels or colors to the edges or vertices of a graph The most common types of graph colorings are edge coloring and vertex coloring. Edge Coloring - an assignment of labels or colors to each edge of a graph such that adjacent edges (or the edges bounding different regions) must receive different colors. Vertex Coloring - an assignment of labels or colors to each vertex of a graph such that no edge connects two identically colored vertices

One of the reasons that graph coloring became a subfield of graph theory and that graph theory became a its own branch of math is because of the well-known Four Color Problem

The Four Color Problem The four color problem states that any map in a plane can be colored using four-colors in such a way that regions sharing a common boundary (other than a single point) do not share the same color. 

The History The Four Color Problem came about when Francis Guthrie was working with coloring a map of England and discovered that he could color the map using only 4 colors . Francis was trying to show that any map can be colored using only four colors so that no regions sharing a common share the same color. Francis’s brother (Frederick Guthrie) was a student of De Morgan, and Francis asked his brother to show the maps to him to prove if this is always true.

History Continued… De Morgan studied the problem in 1852 and could not figure out a solution. He wrote to another mathematician (Sir William Rowan Hamilton) about the problem. However, Hamilton felt the problem would not be in his site of study for a while. So De Morgan looked to other mathematicians for help. Many mathematicians worked on it over the years. 1879 - Alfred Bray Kempe announced that he had solved the Four-Color Problem. 1890 - Percy John Heawood proved Kempe’s proof to be incorrect, and then proved that every map can be colored using 5 colors.

History Continued… Heawood also proved that if the number of edges around each region is divisible by 3 then the regions are 4-colorable. George Birkhoff then worked on the problem for many years, but his work ultimately led Phillip Franklin to prove that the four color problem is true for maps with at least 25 regions.

Potential Solution 1976 - Kenneth Appel and Wolfgang Haken used a computer to prove the Four Color Problem. Using a proof by contradiction, they assumed a map that needed more than 4 colors and then developed a formula to look at maps using numbers and algorithms to express neighboring countries. Their program looked at almost 2,000 maps and proved that it must be true since no same colored region appeared to touch. It took the program almost 1,200 hours to receive the results.

A Solution? While many believe this is not true, no one has been able to disprove it. Many do not accept this proof because it uses a computer, but the math involved is too tedious and difficult to prove by hand. As a result of the Four Color Problem, other areas of research have opened up to mathematicians.

Current Research Determining the chromatic number of a graph The chromatic number of a graph is the least number of colors required to do a vertex-coloring of a graph Calculating the chromatic number of a graph is an NP-complete problem  "no convenient method is known for determining the chromatic number of an arbitrary graph"

Examples

References Duke, M., Wagner, A., Holman, J., & Morris, E. (2012). The Four Color Problem. PowerPoint Presentation presented during Fall semester 2012. http://www.math.niu.edu/~rusin/known- math/index/05CXX.html http://mathworld.wolfram.com/ChromaticNumber.html http://mathworld.wolfram.com/Graph.html http://www.britannica.com/EBchecked/topic/242012/graph- theory http://www-groups.dcs.st- and.ac.uk/~history/HistTopics/The_four_colour_theorem.ht ml http://mathworld.wolfram.com/Four-ColorTheorem.html