Graph Theory and Graph
Coloring
Lindsay Mullen
•
(Abstract) Algebra and Number Theory
• Combinatorics (Discrete Mathematics)
• Graph Theory
• Graph Coloring
What is Graph Theory?
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Branch of mathematics concerned with networks of
points connected by lines
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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.
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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
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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.
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In 1735 the Swiss mathematician Leonhard Euler
presented a solution to this problem, concluding that
such a walk was impossible.
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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.
Graphs
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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
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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
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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
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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
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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.
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History Continued…
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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
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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.
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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"
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Examples
References
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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/knownmath/index/05CXX.html
http://mathworld.wolfram.com/ChromaticNumber.html
http://mathworld.wolfram.com/Graph.html
http://www.britannica.com/EBchecked/topic/242012/graphtheory
http://www-groups.dcs.stand.ac.uk/~history/HistTopics/The_four_colour_theorem.ht
ml
http://mathworld.wolfram.com/Four-ColorTheorem.html