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
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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.
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start
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finish
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Assignment
• Worksheet Euler Circuits and eulerization