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GRAPH THEORY

- By Jen Willig

Outline

- What is graph theory?
- Leonard Euler
- Different types of graphs
- Graph models
- Two specific
- Traveling salesperson problem
- Map coloring problem

- Graph theory was brought about to study methods

of making a path from one point to another

without retracing any steps - VERTEX the point at which two or more sides

meet. - ODD NUMBER a number not divisible by 2

Leonard Euler

- April 15, 1707 Born
- Introduced graph theory in the 18th century
- Konigsberg bridge problem

Bridge Problem

- Problem Wondering whether or not one could walk

around the city in a way that would involve

crossing each bridge exactly once.

Bridge Solution

- Euler realized he could not do this without

crossing at least one bridge twice - Solution He concluded that any problem of this

sort must be represented by replacing areas of

land by points and bridges with arcs

- Euler then stated that there is no possible way

to go from one point to another by arc without

going back to a point at least once and being

stuck with no place to go.

- Conclusion Every vertex with an odd number of

arcs attached has to be either a beginning or an

end and there can be only up to 2 odd vertices

Terms

- Edge any line drawn from one dot to another
- Vertex Each dot that appears in a collaboration

of dots - Degree (of a vertex) The number of edges that

touch a vertex - Size of a graph is the number of vertices that

the graph has

- Regular type of graph with every vertex having

the same degree - Path the route traveled along edges and through

vertices - Cycle path that begins and ends at the same

vertex - Node connecting point at which several lines

come together

Graph Theory

- A graph is a collection of dots that may or may

not be connected by lines. - It doesnt matter how the dots are connected as

long as 2 dots are only connected with one line.

Graphs

- Graphs are structures that contain vertices that

are all connected. - All graphs are determined depending on the number

of edges that are connected by the vertices.

Simple Graph

- Thought of as G (V,E)
- V is a non empty set of vertices
- E is a set of unordered pairs of distinct

elements of V called edges

- Example of a simple graph
- Computer Network consisting of computers and

telephone lines between each computer - Each computer is represented by a point

- Each phone line is represented by an edge
- IF each computer is only connected by one phone

line then the network is represented as a simple

graph

Multigraphs

- Not every multigraph is a simple but every simple

graph can be a multigraph - Reasoning there can be two or more edges that

connect the same set of vertices

- Similar to a simple graph but it more arcs from

one vertex to another - In this case multiple edges are allowed

Pseudograph

- Another branch off the simple graph
- Has a loop
- Loop a single edges going from one vertex to

itself

- The most general type of undirected graph
- They contain many loops and multiple edges

Pseudograph

Directed and Undirected Multigraphs

- Directed graph can have multiple edges and

loops, but the direction must be different for

each edge

- Undirected Graph has multiple edges that can go

in the same direction

Undirected Graph

Directed Graph

Graph Models

- Niche overlap graphs
- Influence graphs
- Round robin graphs

Niche overlap graphs

- Found when dealing with animals
- Many overlapping areas, such as food webs.
- When certain animals breed with other animals to

produce a new species - Each animal is represented as a vertex and they

are connected by undirected edges

Influence Graph

- Identifies power and influence structure
- Umbrella example
- Should you take an umbrella or should you leave

the umbrella at home?

Umbrella Graph

Round Robin Graphs

- Known in sports
- Example 6 team tournament
- Each vertex represents each team and goes to

every other vertex in the graph. - Each edge represents each of the teams playing

each other

Special Simple Graphs

- Special simple graphs are complete graphs because

there is one edge between each pair of vertices

Cycles

- Special simple graphs that contain cycles are the

basic geometric figures - The number of cycles in each figure depends on

the number of vertices and edges. - In determining if a graph has a cycle, there must

be no edges on the inner part of the figure.

Triangle

Square

Pentagon

Wheels

- Special simple graphs that contain wheels are

graphs that have a single point in the middle of

the figure and there are edges going from each

outer vertex to the single inner vertex

WHEEL GRAPHS

TRAVELING SALESPERSON PROBLEM

- One of the easiest to explain but one of the most

difficult to solve - Problem states A traveling salesperson needs to

visit a certain number of cities before returning

home

TRAVELING SALESPERSON PROBLEM

- The salesperson wants to know what route can be

taken so every city can be visited without

retracing any steps and returning back home - So now. . . .

TRAVELING SALESPERSON PROBLEM

- ACTIVITY!
- YOU are the traveling salesperson ?
- Everyone starts in City A.
- Must end up back in City A.
- How many different paths can there potentially be?

What is your shortest route?

- A to B 10 miles
- A to C 3 miles
- A to D 6 miles
- A to E 3 miles
- C to D 4 miles
- C to E 8 miles

- B to C 3 miles
- B to D 5 miles
- B to E 2 miles
- D to E 3 miles

- The Odyssey
- Homer wanted to visit all 16 cities.
- A team of mathematicians figured out that total

possibilities to visit each city without

retracing steps were - 653, 837,184,000

TRAVELING SALESPERSON PROBLEM

- When a salesperson goes from each city from

another directly then the graph is considered to

be complete. - When a salesperson returns back home then the

trip is said to be a round trip. - The length of the trip is the sum of the lengths

of the lines in a round trip.

Graph Coloring

- As a child, remember when coloring you may not

have wanted touching sides to be colored the

same? - That is graph coloring
- http//www.c3.lanl.gov/mega-math/workbk/map/mpprst

ory.html

Graph Coloring

- Idea was to assign each vertex a color.
- It was said that the number of areas could be

colored using a less amount of colors than

vertices. - GOAL Not to color adjoining sides the same color

MAP COLORING PROBLEM

- Assertion made early on that only 4 colors were

needed in coloring any map - Just an assumption because there were no maps

that had ever needed 5 colors - Posed a problem for mathematicians

MAP COLORING PROBLEM

- Dates back to October 23, 1852
- Francis and FredrickGuthrie
- University College London
- Questions about Map Coloring to Professor

DeMorgan - Sent the problem off to Colleagues

MAP COLORING PROBLEM

- Solution came back to DeMorgan two decades later

on July 17, 1879 - Alfred Bay Kempe claimed he had a proof to the

Four Color Conjecture

FOUR COLOR THEOREM

- Proof If we have a map in which every region is

colored red, green, blue, or yellow except one,

X. If this final region X is not surrounded by

regions of all four colors then there is a color

left for X. Hence suppose that regions A, B, C,

D in order, colored red, yellow, green, and blue

then there are two cases

- Case 1 There is no chain of adjacent regions

from A to C alternately colored red and green - If this conjecture holds true then there is no

problem, just change A to green and interchange

the number of red/green regions adjoining A.

Since C is not in the chain, it remains green and

there is now no red region adjacent to X,

therefore X can be red

- Case 2 There is a chain of adjacent regions

from A to C alternately colored red and green - If this case holds true there can be no chain of

yellow/blue adjacent regions from B to D. Hence

case 1 holds true for B to D and we change colors

as already stated.

FOUR COLOR CONJECTURE

- In 1890, The Four Color Theorem became the Four

Color conjecture - Percy John Heatwood showed Kempes proof to be

wrong - 60 years of his life working on the map coloring

problem - Added to his conjecture in 1898 that if the

number of edges around each region is divisible

by three then the regions are four colorable.

GRAPH COLORING

- After a while it was stated that it was ok if two

alike vertices are the same color as long as

their edges are not the same color - 1922 it was stated that only maps with 25

regions or less were four colorable

Graph Coloring

- Numbers increased as time moved on
- The largest number of regions in a map that is

four colorable is one with 95 regions which was

discovered in 1972 by a man named Mayar

Graph Coloring

- 1976 the Four Coloring Conjecture became the Four

Coloring Theorem for the 2nd and last time. - Final proof came from two men, Apel and Haken.
- The Four Color Theorem was the first major

theorem proved by a computer that was not able to

be proved directly by a mathematician.

CONCLUSION

- Graph theory has been around us since our very

early childhood, from coloring pictures, to

geometry, to playing in an athletic tournament,

to figuring out the best possible route on a

trip. - Everywhere around us there are graphs and graph

theory.

YOUR TURN!

- Can you draw any of these pictures using only 4

colors?

- THE END!

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