You may want to download the lecture slides that were used for these videos (PDF).
1. Induced Subgraphs & Cut Vertices
We introduce two concepts, induced subgraphs and cut vertices, and provide examples. (5:08)
2. Special Classes of Graphs
This video defines and provides a few examples of special classes of graphs (cycles, complete graphs, cliques, trees). (3:03)
3. Properties of Trees
This video defines leaves, and proves that every tree with at least 2 vertices has at least two leaves. (4:37)
4. Counting Unlabelled Trees
Can you explain why the 6 unlabelled trees on 6 vertices are the ones shown at the start of the video? (2:56)
5. Counting Trees, Continued
In this video we look at counting unlabelled and labelled trees. (11:50)
6. Trails & Circuits in Graphs
In this video we define trails, circuits, and Euler circuits. (6:33)
7. Euler’s Theorem
In this short video we state exactly when a graph has an Euler circuit. (0:50)
8. Algorithm for Euler Circuits
We state an Algorithm for Euler circuits, and explain how it works. (8:00)
9. Why the Algorithm Works, & Data Structures
Here, we discuss why the algorithm for Euler circuits works. Then, we discuss how the way a graph is stored computationally can affect the algorithm for Euler circuits. (12:30)
10. Hamiltonian Paths & Cycles
Here, we return to discussing Hamiltonian paths and cycles, comparing them to Eulerian paths and circuits. (5:48)
11. Maximum Clique Size & Graph Coloring
In this video we introduce two new questions about graphs. First, given a graph, can we find the largest clique it contains as an induced subgraph? Finding the maximum clique size is very hard. Then, we introduce the graph coloring problem, through a situation where we need to store chemicals in a set of rooms. (12:12)
12. Comparing Clique Number to Chromatic Number
After a few technical difficulties, Professor Trotter introduces the inequality Χ(G) ≥ ω(G). Then, he shows that this statement is not an equality in general by looking at a 5-cycle.