We can decide whether or not $G$ is planar.

What's left in this Unit?

  • Today: Drawing graphs on other surfaces
  • Friday: Euler's Theorem and applications
  • Final unit: Colouring Graphs
  • Revision+extras!

    • A video game world:

    Surfaces

    A surface is a space that "locally looks like" $\mathbb{R}^2$

    • $\mathbb{R}^2$
    • Sphere $S^2$
    • Torus $T$
    • Mobius band $M$
    • Klein bottle $K$
    • ...

    Motivation:

    • Drawing graphs on the torus arises naturally
    • Euler's theorem will let us answer the following question:

    Couldn't the video game designers do better?

    Follow the video game example

    To draw a graph on the torus, draw it inside a square

    If an edge hits a side of the square, it comes back at the same spot on opposite edge

    Can we draw $K_5$ on the torus? How about $K_6$? $K_n$?

    Mobius band:

    Like cylinder, but glued with a half twist

    • Can represent Mobius band in plane...
    • Can you draw $K_5$ on the Mobius band? $K_n$?
    • What happens when you cut a Mobius band in half?

    Other surfaces?