Graph Theory Summer Math Workshop for High School

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Graph TheorySummer Math Workshop for High
School Middle School Teachers Sarah
HollidayJune 12, 2009
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Graph theory is not...

• Cartesian plotting

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Graph theory is structure and relationships

• Alice and Bob take geometry together
• Alice and Chuck take French together
• Bob and Dave take history together
• Chuck and Ed take English together
• Bob and Ed take Latin together
• Dave and Ed take PE together

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Graph theory is structure and relationships
A
B
C
E
D
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Graph theory is not...

• Just pictures

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Graph theory includes

• Colouring
• Planarity
• Isomorphisms
• Counting
• Coding
• Designs

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Graph theory is problem solving

• Unlike an algebraic question in which a specific
  numeric answer is sought, a graph theory problem
  will more often ask for a sentence. The results
  from graph theory problems are abstract provide a
  launching point for writing across the curriculum
  and group discussions

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Colouring

• Colouring a graph is the process of assigning
  labels to the vertices.
• A proper colouring is one in which no two
  adjacent vertices have the same colour or
  assigned label.
• A minimum proper colouring is a proper colouring
  using the fewest possible colours.

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Colouring
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A colouring
2
2
1
3
5
2
3
3
4
2
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A proper colouring
2
2
1
3
5
2
5
3
4
2
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A minimum proper colouring
2
2
1
3
1
2
1
3
4
2
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A colouring problem

• A department chair needs to schedule nine classes
  A through I in as few rooms as possible. There
  are certain conflicts that must be avoided. How
  can the chair schedule the classes?

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A colouring problem

• Sue teaches classes A, B, C, and D.
• Tim teaches classes E, F, and G.
• Una teaches classes H and I.
• Classes A, F, and I use the projector.
• Classes B, E, G, and H use the legos.

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A colouring problem
A
B
C
I
D
H
E
G
F
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A colouring problem
1
2
A
B
3
C
3
I
4
D
H
4
E
1
G
F
3
2
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A colouring problem

• One solution
• In time slot 1, classes A and E
• In time slot 2, classes B and F
• In time slot 3, classes C, G and I
• In time slot 4, classes D and H

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Colouring maps

• We can use a graph to represent physical
  relationships. We will mark a vertex for each
  state, and draw an edge between two vertices that
  share a border (not just a corner).

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Colouring maps
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Colouring maps
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Colouring maps

• How many colours are required to properly colour
  a map or a graph that represents a map?
• The second example shows us that four colours
  suffice.
• A graph that has a map representation is called
  planar.

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Planarity

• A graph that can be represented using vertices
  and edges that do not cross is planar we say
  that the graph can be drawn in the plane.

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Planarity
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Planarity
B
A
C
D
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Planarity
B
A
C
D
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Planarity
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Planarity
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Planarity
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Planarity Euler's formula

• The problem of determining which graphs are
  planar was indirectly addressed by Euler in the
  formula V-EF2, where V is the number of
  vertices, E edges and F faces.

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Isomorphisms

• Two graphs that have the same relationships but
  possibly different drawings are called
  isomorphic.
• http//www.planarity.net/

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Counting

• One family of graphs is called the complete
  graphs, or cliques. These graphs include all
  possible edges between each pair of vertices.
• An interesting exercise that illustrates some of
  the fun of graph theory is to count the number of
  edges in a clique on n vertices.

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Counting
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Counting
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Counting
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Counting

• The number of edges in the (n1)-clique is the
  number of edges in the (n)-clique n.

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Counting

• There are n vertices. Each vertex is adjacent to
  all the others. There are (n-1) other vertices,
  so that makes n(n-1) edge-ends. Each edge has
  two ends, so there are n(n-1)/2 edges in an
  n-clique.

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Coding
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Coding
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Coding
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Coding
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Coding
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Coding
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Coding
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Coding
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Coding
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Coding
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Coding
0
1
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Coding
00
01
10
11
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Coding
000
001
101
010
110
111
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Coding
0000
0001
1001
1101
0010
0110
1111
1110
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Coding

• Error-detecting codes will be made of a
  dictionary of words that are nonadjacent vertex
  labels
• 0
• 01,10
• 101, 000, 110, 011

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Designs

• The design of experiments in this context is the
  search for an efficient distribution of trials to
  eliminate redundancies while still providing
  comparison data.

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Designs

• Acme cookie company has seven varieties of cookie
  to test Almond, Butter Chocolate, Date,
  Everything, Fig and Gingersnaps. Their oven
  holds three cookie trays at a time.

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Designs
A
B
G
C
F
D
E
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Designs
A
B
G
C
F
D
E
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Designs
A
B
G
C
F
D
E
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Designs
A
B
G
C
F
D
E
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Designs
A
B
G
C
F
D
E
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Designs
A
B
G
C
F
D
E
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Designs
A
B
G
C
F
D
E
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Further topics

• Decompositions
• Domination
• Extrema
• Topology