Lecture Videos Lecture 10 – Perfect Graphs, Interval Graphs, & Coloring Algorithms

Lecture 10 – Perfect Graphs, Interval Graphs, & Coloring Algorithms

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)