You may want to download the lecture slides that were used for these videos(PDF).
1. Chromatic Number & Girth
In this video we review concepts introduced in previous lectures on chromatic number and girth, and discuss a classic result by Erdős. (4:12)
2. Perfect Graphs
In this video we introduce perfect graphs. A graph is not perfect if it contains an odd cycle on 5 or more vertices as an induced subgraph. (7:16)
3. The Complement of Graph
This video introduces the definition of a complement of a graph. A graph is not perfect if its complement contains an odd cycle on 5 or more vertices as an induced subgraph. (11:10)
4. Berge’s Perfect Graph Conjecture
This video reviews the history of an important conjecture (later proved) that was made in 1961: A graph is perfect if and only if neither the graph nor its complement contains an odd cycle with at least five vertices as an induced subgraph. (10:21)
5. Intersection Graphs
This video introduces the definition of an intersection graph. Every graph can be written as an intersection graph. Another example of intersection graphs is discussed. (8:52)
6. Interval Graphs
This video discusses an application of some of the ideas we’ve explored to interval graphs. The First Fit algorithm is introduced. (3:33)
7. How First Fit Works
This video explains how the First Fit algorithm works. For many graphs, the First Fit algorithm does not find an optimal coloring. (6:26)
8. Coloring as a Two-Player Game
Let’s think of graph coloring as a game with two people: one person builds the graph, the other person colors the graph. Can the builder force the colorer to use too many colors? (13:17)
9. Adversarial Algorithms & Non-Optimal Coloring
This video gives a discussion of a real-world application of graph theory related to databases and errors in them. (2:30)
10. Interval Graphs & First Fit
Let’s explain why First Fit coloring is optimal for interval graphs. The case k = 1 is obvious, the case for k > 1 is more subtle. (3:04)
11. A Theorem by Kierstead & WTT
In the last few minutes of this lecture, WTT introduces a theorem that states there is a strategy for coloring an unknown interval graph online, where if the largest clique built has size ω, the number of colors used will be no more than 3ω – 2. Here, “unknown” means that the each vertex in the graph is revealed as it is colored. (5:10)