Lecture 12 – More on Coloring & Planarity

You may want to download lecture slides that were used for these videos (PDF).

1. Review of Planar Graphs

We start this lecture with rephrasing concepts from the previous lecture, including planar graphs, Euler’s Theorem, homeomorphs, and the maximum number of edges in a planar graph. (3:25)

2. Two-Colorable Planar Graphs

This video offers a set of theorems that have to do with two-colorable planar graphs. WTT describes how we would prove that a 2-colorable planar graph must have q < 2n – 4, which in turn implies that the complete bipartite graph K3,3 is non-planar.  Then, we present Kuratowski’s Theorem, and the Four Color Theorem. (8:46)

3. Coloring Planar Maps

We go back to the map of Georgia that we introduced in an earlier lecture that has 5 colors. Mr. Burr could have done better. WTT also gives us some of the history and a few stories on the Four Color Theorem. (5:47)

4. Planar Graphs & Planar Maps

We connect planar graphs and planar maps by discussing the dual graph of a graph. (3:26)

5. The Four Color Theorem

This video gives more details on the history of the proof of the four color theorem. (13:03)

6. Game Coloring for Graphs (1)

This video introduces the game chromatic number, where two players take turn coloring the graph.  Then, we look at an example. (6:05)

7. Game Coloring for Graphs (2)

Another example of game coloring. (3:17)

8. Bounding the Game Chromatic Number of a Planar Graph

An area of ongoing research is focused on refining the upper and lower bounds on the game chromatic number for a planar graph. (4:56)

9. The Game Chromatic Number of a Tree

WTT gives an exercise for you to try: finding the game chromatic number of a tree. (3:54)

10. List Coloring

List coloring is introduced, which is something that can be much harder than ordinary coloring. (4:24)

11. (Thomasen, 1994) The List Chromatic Number is at most 5

In this brief video we state a theorem that gives an upper bound for the list chromatic number. (1:23)