Lecture 7 – More Graph Theory Basics: Trees & Euler Circuits

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.