Planar Graphs, Solids, and Surfaces
11 April 2014
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Last time we discussed the Four Color Theorem, which says that any
map can be colored with at most 4 colors and not have two regions
that share a border having the same color.
It turns out that this result has something to do with the topology of
the plane. If we draw maps on other surfaces, we may need a
different number of colors.
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For example, if we draw a map on a torus (a doughnut), we may need
up to 7 colors, as the following pictures indicate.
http://faculty.smcm.edu/sgoldstine/torus7.html
Here is an animation of a map on the torus which needs 7 colors.
The map on the torus shown above has 7 regions and each region is
adjacent to each of the other 6 regions. This is why it needs 7 colors.
The next video really doesn’t have any mathematics in it, but it
shows an important application of the torus.
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While it may seem that determining the minimum number of colors
to color a map drawn on a torus would be much harder than for maps
drawn on a plane, it is just the opposite.
The Four Color Theorem for maps on a plane took a huge amount of
effort, 61 pages in Scientific American, and lots of computer
calculation, the result for the torus can be written down in a few
pages, and was proved much earlier.
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Planar Graphs
The graphs that arise from a map turn out to have the following
property: They can be drawn in such a way that no two edges cross.
These are called planar graphs.
The pentagon graph we saw the previous class is not a planar graph.
This is why it does not represent the graph of a map and why it does
not violate the Four Color Theorem.
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Let’s consider the following graph with 4 vertices and 6 edges:
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Clicker Question
Can we redraw the graph so that the edges do not cross?
A Yes
B No
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Answer
Yes we can redraw it by having one of the diagonal edges drawn
outside of the square. Thus, it is a planar graph, even though the
original drawing does not indicate so.
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Faces of a Planar Graph
For solids, such as those above, there is a reasonable meaning of
vertex, edge, and face. For example, the cube has 8 vertices (or
corners), 12 edges, and 6 square faces.
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The cube can be drawn as a graph in multiple ways. Here are two
such. One is a pretty typical way to draw a cube, and the other is less
so.
The second graph is a planar graph. One way to view the six faces of
the cube in it are to consider the 5 regions inside the graph along
with the outside.
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In general, with a planar graph, we can see regions inside the graph,
and we say the number of faces of the graph is the number of regions
inside plus the outside. In this way a graph that represents a solid will
have the same number of faces as the solid.
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Clicker Question
How many faces does the Octahedron have?
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Answer
There are 8 faces. Looking at the figure on the left, there are 4 faces on
the top half, and 4 on the bottom half.
Looking at the graph on the right, there are 7 regions inside the graph,
and 1 outside, making 8 faces.
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Is there a relationship between the Numbers of Vertices,
Edges, and Faces?
V
4
8
6
20
12
E
6
12
12
30
30
F
4
6
8
12
20
Solid
Tetrahedron
Square
Octahedron
Dodecahedron
Icosahedron
Stare at this table for a bit and see if you can find any relationship
between V , E , F that holds for all of the solids.
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Euler’s Formula
One of Euler’s other contributions to graph theory was the following
result about planar graphs:
If V is the number of vertices, E the number of edges, and F the
number of regions formed by a planar graph, then
V −E +F =2
This formula was a key factor in the proof of the Four Color Theorem.
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Platonic Solids
The solids in this picture are called Platonic solids, named after the Greek
philosopher Plato. These solids, the most regular solids that are built from
plane figures, were given nearly mystical importance by the ancient Greeks.
While they were aware of these five shapes, they didn’t know if there were
any others.
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How Many Platonic Solids Are There?
Euler’s formula can help us determine all Platonic solids. The
distinguishing characteristics of these solids are:
• Each face is a regular n-gon for some n. This means each face has n
sides, each side is the same length, and each interior angle is the
same. Examples of these faces are equilateral triangles, squares,
regular pentagons, and regular hexagons.
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• Each vertex is connected to the same number of edges.
• The same number of faces touch each vertex.
We will see that Euler’s formula can be used to show that the 5
Platonic solids are the only possible.
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Let’s consider an unspecified Platonic solid. Call n the number of
edges surrounding each face. If F is the number of faces, then is nF
the number of edges?
Not quite. Since each edge is shared between two faces, this double
counts the number of edges. So, E = nF /2.
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How many vertices are there? Let c be the number of edges coming
together at each vertex. Is cV the number of edges?
Not quite; this double counts the number of edges. Since each edge
hits exactly two vertices, we have E = cV /2.
Solving for F and V gives F = 2E /n and V = 2E /c.
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Putting the formulas F = 2E /n and V = 2E /c into Euler’s formula
V −E +F =2
gives
2E
2E
−E +
=2
c
n
We can simplify this as
E
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2
2
−1+
c
n
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In order to have a solid, each face must be surrounded by at least 3
edges. So, n ≥ 3.
Also, we must have at least 3 faces attached to each vertex. So,
c ≥ 3.
The value of E is a positive number.
2
−1+
E
c
this forces
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Because
2
=2
n
2
2
−1+ >0
c
n
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The inequality
2
2
−1+ >0
c
n
gives a serious restriction on the values of n and c.
With some algebra we can rewrite the inequality as
1
1 1
+ >
c
n
2
or
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1
1 1
> −
c
2 n
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Since n ≥ 3, we see that 1/n ≤ 1/3. So,
1 1
1
> −
c
2 n
1 1
1
> − =
2 3
6
This forces c < 6. Thus, c can only be 3, 4, 5.
For example, if c = 3, then we get
1 1
1
+ >
3 n
2
This forces 1/n > 1/6, so n < 6. So n can be 3, 4, 5.
Similar reasoning only produces two more cases, n = 3 and c = 4, 5.
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The table below represents all possible values of c and n which will
allow Euler’s formula to be satisfied.
c
3
3
3
4
5
n
3
4
5
3
3
V
4
8
20
6
12
E
6
12
30
12
30
F
4
6
12
8
20
Solid
Tetrahedron
Square
Dodecahedron
Octahedron
Icosahedron
This shows that the 5 Platonic solids known to the ancient Greeks are
all that can exist.
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Houses and Utilities Problem
Suppose there are 3 utilities and 3 houses. Each house is to be
connected to each utility (by, e.g., a pipe, or wire). Is it possible to do
this without having the connections crossed?
In this picture, think about the utilities as the vertices in red and the
houses in blue.
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Clicker Question
Can you see a way to connect each reg vertex with each blue vertex
without having any lines crossed?
A Yes
B No
C Not sure
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Here is an attempt to solve the problem. It doesn’t quite work. Why not?
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Here is another attempt to solve the problem. It also doesn’t quite work.
Why not?
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It turns out that no matter how hard you try, it is not possible to
draw this graph without crossing edges.
Euler’s formula is actually a result about the nature of surfaces. We’ll
investigate this further next week.
Planar graphs mean something different if you draw them on a
surface other than a piece of paper.
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For example, we can solve the utilities problem if the houses and
utilities were drawn on a torus (a doughnut). The following animation
shows how each house can be connected to each utility without
having the lines cross, providing we do this on the torus. What this
means is that what it means to be a planar graph depends on what
surface do we draw the graph.
The website
http://lsusmath.rickmabry.org/rmabry/live3d/k33-torus.htm
gives an animation of this graph.
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Next Time
We will look further at different surfaces, including the plane and the
torus, but consider others. We’ll see that Euler’s formula says something
about surfaces. We’ll also look at other properties of surfaces, and see
that the properties we discuss are enough to classify all surfaces.
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