Vertex-Edge Graphs

Euler Paths

Euler Circuits

Leonard Euler

• This problem is an 18th century problem that intrigued Swiss mathematician Leonard Euler (1707-1783).

• This problem was posed by the residents of Königsberg, a city in what was then Prussia but is now Kaliningrad.

The Seven Bridges of Konigsberg

Euler (pronounced “oiler”) Paths

• Vocabulary

• Theory

• Problem and Story

• Examples and Non-Examples

• Try Some Puzzles

Vocabulary• Vertex: point (plural-vertices)• Edge: segment or curve connecting the vertices• Odd vertex: a vertex with an odd number of edges

leading to it• Even vertex: a vertex with an even number of

edges leading to it• Euler path: a continuous path connecting all

vertices that passes through every edge exactly once

• Euler circuit: an Euler path that starts and ends at the same vertex

Vertex

• Point• Name the points

(vertices) on this vertex-edge graph

Edges• segment or curve

connecting the vertices

• What are the edges on this vertex-edge graph?

Odd Vertex

• a vertex with an odd number of edges leading to it

• What are the odd vertices on this vertex-edge graph?

Even Vertex

• a vertex with an even number of vertices leading to it

• What are the even vertices on this vertex-edge graph?

Königsberg Bridges

• In the 1700s, seven bridges connected two islands on the Pregel River to the rest of the city.

• The people wondered whether it would be possible to walk through the city by crossing each bridge exactly once and return to the original starting point.

Euler’s Solution

• Using a graph like the picture where the vertices represent the landmasses of the city and the edges represent the bridges, Euler was able to find that the desired walk throughout the city was not possible.

• In doing so, he also discovered a solution to problems of this general type.

Euler’s Solution (cont’d)

• Euler found that the key to the solution was related to the degrees of the vertices.

• Recall that the degree of a vertex is the number of the edges that have the vertex as an endpoint.

• Find the degree of each vertex of the graphs on the previous slide. Do you see what Euler noticed?

Euler’s Solution

• Euler found that in order to be able to transverse each edge of a connected graph exactly once and to end at the starting vertex, the degree of each vertex of the graph must be even. (As only in the second graph)

Euler Circuits and Paths

• In his honor, a path that uses each edge of a graph exactly once and ends at the starting vertex is called an Euler circuit.

• Euler also noticed that if a connected graph had exactly two odd vertices, it was possible to use each edge of the graph exactly once but to end at a vertex different from the starting vertex. Such a path is called an Euler path.

Euler Path

• a continuous path connecting all vertices that passes through every edge exactly once

• Is this vertex-edge graph an Euler path?

• Why, or why not?

Euler Circuit

• an Euler path that starts and ends at the same vertex

• Is this vertex-edge graph an Euler circuit?

• Why, or why not?

Another Example

• Is this vertex-edge graph an Euler path?

• Why or why not?

• Is this vertex-edge graph an Euler circuit?

• Why, or why not?

Euler’s Theorem, or Rules•If a graph has more than two odd

vertices, then it does not have an Euler path.

• If a graph has two or fewer odd vertices, then it has at least one Euler path.

• If a graph has any odd vertices, then it cannot have an Euler circuit.

• If every vertex in a graph is even, then it has at least one Euler circuit.

Let’s Take Another Look

• How many odd vertices in this vertex-edge graph?

• According to Euler’s Theorem, can this be an Euler path?

• Can it be an Euler circuit?

Let’s Look at the Other One

• How many odd vertices in this vertex-edge graph?

• According to Euler’s Theorem, can this be an Euler path?

• Can it be an Euler circuit?

Pencil Drawing Problem-Euler Paths

Which of the following pictures can be drawn on paper without ever lifting the pencil and without retracing over any segment?

Pencil Drawing Problem-Euler Paths

Graph Theoretically: Which of the following graphs has an Euler path? First, identify the points.

Pencil Drawing Problem-Euler Paths

Answer: the left but not the right.

start finish

1 2

3

4

5

6

Assignment

• Worksheet Euler Circuits and eulerization