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 K_{3,3} 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)