1. Chapter 10 Graphs and Trees
This handout:
Terminology of Graphs
Eulerian Cycles

2. Terminology of Graphs A graph (or network) consists of a set of points
a set of lines connecting certain pairs of the points.
The points are called nodes (or vertices).
The lines are called arcs (or edges or links).
Example:

3. Graphs in our daily lives
Transportation
Telephone
Computer
Internet
Social networks
Electrical (power)
Pipelines
Molecular structures in biochemistry

4. Terminology of Graphs
Each edge is associated with a set of two nodes, called its endpoints.
Ex: a and b are the two endpoints of edge e
An edge is said to connect its endpoints.
Ex: Edge e connects nodes a and b.
Two nodes that are connected by an edge are called adjacent.
Ex: Nodes a and b are adjacent. a b c e f

5. Terminology of Graphs: Paths
A path between two nodes is a sequence of distinct nodes and edges connecting these nodes.
Example:
Walks are paths that can repeat nodes and arcs. a b

6. A little history: the Bridges of Koenigsberg
“Graph Theory” began in 1736
Leonhard Eüler
Visited Koenigsberg
People wondered whether it is possible to take a walk, end up where you started from, and cross each bridge in Koenigsberg exactly once

7. The Bridges of KoenigsbergA 1 2 3 B 4 C 5 6 7 D
Is it possible to start in A,
cross over each bridge exactly once,
and end up back in A?

8. The Bridges of KoenigsbergA 1 2 3 B 4 C 5 6 7 D
Translation into a graph problem: Land masses are “nodes”.

9. The Bridges of KoenigsbergA 1 2 3 B 4 C 5 6 7 D
Translation into a graph problem : Bridges are “arcs.”

10. The Bridges of KoenigsbergA 1 2 3 B 4 C 5 6 7 D
Is there a “walk” starting at A and ending at A and passing through each arc exactly once? Such a walk is called an eulerian cycle.

11. Adding two bridges creates such a walk3 8 1 2 B 4 C 5 6 9 D 7
Here is the walk.
A, 1, B, 5, D, 6, B, 4, C, 8, A, 3, C, 7, D, 9, B, 2, A
Note: the number of arcs incident to B is twice the number of times that B appears on the walk.

12. Existence of Eulerian Cycle4 1 2 4 3 7 6 5 A C D B 8 9
The degree of a node is the number of incident arcs 6 4 4
Theorem. An undirected graph has an eulerian cycle if and only if (1) every node degree is even and (2) the graph is connected (that is, there is a path from each node to each other node).