Chapter 5
Euler Circuits
Leonhard Euler (1707 - 1783)
..
While visiting the town of Konigsberg, in Eastern Europe,
Euler was posed with a puzzle..
Can a person walk around town in a way that each of the
7 bridges is crossed only once?
So in solving this puzzle, Euler laid the foundations for a new
type of geometry, which developed into one of the most
practical branches of mathematics -graph theory .
Unicursal tracings - tracing a drawing without
lifting the pencil or retracing any of the lines.
Closed Unicursal tracing - when you end in the
same place you started.
Open Unicursal tracing - when you start and end
in different places.
Graph: a picture consisting ofdots, called vertices,
and lines, called edges.
*The edges do not have to be straight lines,
but they must connect two vertices.
A graph is a structure that defines pairwise relationships within
a set of objects. The objects are vertices, and the pairwise
relationships are the edges.
so X is related to Y if and only if XY is an edge.
Loop - An edge connecting a vertex back with itself.
A
Example:
B
E
C
D
In this graph, there are 5 vertices and 6 edges.
The vertex set is V = {A, B, C, D, E}
The edge set is E = {AB, AD, BB, BD, CD, CD}
Notice the edge BB is a loop.
Notice the vertex E is an isolated vertex.
Notice the graph has more than one edge
connecting C and D. The graph is said to have
multiple edges.
A
B
E
C
D
The graph is not defined by the picture.
The graph is defined by its vertex set and edge set.
A
B
C
D
E
Graph Concepts and Terminology
Adjacent Vertices - two vertices that have an edge joining them.
A
B
E
C
D
For Example:
Vertices A and D are adjacent.
Vertex B is adjacent to itself.
Adjacent Edges - two edges that share a common vertex.
A
B
E
C
For Example:
AD and AB are adjacent edges
D
Degree of a vertex - the number of edges at that vertex.
(Loops contributes twice toward the degree.)
A
B
E
C
D
For Example:
deg(A) = 2
deg(B) = 4
deg(C) =
deg(D) =
deg(E) =
Odd and Even Vertices - An odd vertex is a vertex of odd degree;
an even vertex is a vertex of even degree
Path - a sequence of vertices with the property that each vertex in
the sequence is adjacent to the next one. It describes a sequence of
adjacent edges - a trip along the edges of the graph.
*An edge can be a part of a path only once.
Length of the Path - the number of edges in the path.
More Graph Terminology
Circuit - same as a path, but has the requirement that the trip starts
and ends at the same vertex.
A
B
E
C
D
For Example:
A B D A is a circuit of length 3.
This same circuit can be described as
B D A B or as D A B D
Connected graphs - A graph is connected if, given two vertices,
there is a path joining them. A graph that is not connected is
called disconnected. A disconnected graph is made up of
separate connected components.
A B
C
D
This is a disconnected graph
E
F
G H
A B
C
D
E
G H
This is a connected graph
F
Bridges - An edge that, if erased, the graph would become
disconnected.
A B
E
F
C
D
G H
For Example:
CH is a bridge
Euler Path - A path that passes through every edge of a graph.
Edges can only be traveled once.
Note: a disconnected graph cannot have a Euler path
Euler Circuit - A circuit that passes through every edge of a
graph.
*An Euler Circuit is a closed unicursal tracing of the edges of
the graph.
* Having a Euler circuit and having an Euler path are
mutually exclusive - if a graph has an Euler circuit it cannot
have an Euler path, and vice versa.