Chapter 6: Graphs

6.2 The Euler Characteristic

Draw A Graph!

• Any connected graph you want, but don’t make it too simple or too crazy complicated

• Only rule: No edges can cross (unless there’s a vertex where they’re crossing)

OK: Not OK:

Now Count on Your Graph

• Number of Vertices: V = ?

• Number of Edges E = ?• Number of Regions (including the region

outside your graph)R = ?

V-E+R: The Euler Characteristic

ANY connected graph drawn on a flat plane with no edge crossings will have the same value for V-E+R: it will always be 2. (We’ll talk about why in a minute.)

V-E+R: The Euler Characteristic

ANY connected graph drawn on a flat plane with no edge crossings will have the same value for V-E+R: it will always be 2. (We’ll talk about why in a minute.)

The value of V-E+R for a surface is called its Euler Characteristic, so the Euler Characteristic for the plane is 2.

V-E+R: The Euler Characteristic

The Euler Characteristic is different on different surfaces. More on this later.

For now, we’re going to stick with graphs on a flat plane.

Why is V-E+R=2 on a flat plane?

Start with simplest possible graph, count V-E+R:

Now, to draw any connected graph at all, you can do it by just adding to this in 2 different ways, over and over.

Adding an Edge but no Vertex

• How does this change V? E? R?

• How does this change V-E+R?

Adding an Edge to a new Vertex

• How does this change V? E? R?

• How does this change V-E+R?

So…

…we can draw ANY connected graph on a flat plane by starting with the basic one-edge, two-vertex graph and building it up step by step.

So…

…we can draw ANY connected graph on a flat plane by starting with the basic one-edge, two-vertex graph and building it up step by step.…the starting graph has V-E+R=2, and each step keeps that unchanged.

So…

…we can draw ANY connected graph on a flat plane by starting with the basic one-edge, two-vertex graph and building it up step by step.…the starting graph has V-E+R=2, and each step keeps that unchanged.…therefore, whatever graph we end up with still has V-E+R=2!

Other Surfaces: Spheres

• Think of graph drawn on a balloon. Then flatten it out:

• Same V, E, R, so same Euler Characteristic!

Other Surfaces: Torus

But some surfaces have different Euler Characteristics, for example a torus (donut):

The Euler Characteristic of a torus is 0, not 2.

Application to 3-D Solid Shapes

We can think of “inflating” a polyhedron with colored edges and corners until it looks like a graph on a sphere:

Th This comes from a cube.

Application to 3-D Solid Shapes

• and for a polyhedron are the same thing as V, E, and R for a graph on a sphere, so we know that

For any polyhedron with no holes in it,

Application to 3-D Solid Shapes

This lets us finally see why there are only 5 regular polyhedra!

Application to 3-D Solid Shapes

Remember that in any regular polyhedron, every face has the same # of edges, which we’ll call

= # of edges / face

Also every vertex has the same # of edges attached to it, so we’ll call that number

= # of edges / vertex

Application to 3-D Solid Shapes

Also remember that and

So and

The Euler Characteristic equation turns into

Application to 3-D Solid Shapes

But we know is positive, so

But also and must be 3 or bigger, and whole numbers

Application to 3-D Solid Shapes

Turns out there are only 5 possibilities for and , and they lead to the 5 regular solids we know about already. So those are the only possible ones!