the bridges of

1
the bridges of Konigsburg
Pamela Leutwyler
2
A river flows through the town of Konigsburg. 7
bridges connect the 4 land masses. While taking
their Sunday stroll, the people of Konigsburg
amused themselves by trying to cross each bridge
EXACTLY ONCE.
Eventually they became frustrated, and they
sent for the mathematician Euler to explain to
them why they were unable to do this. Network
theory was invented.
3
Interpret the problem with a GRAPH
Each land mass is represented by a point called a
vertex
A
B
C
D
4
Interpret the problem with a GRAPH
Each land mass is represented by a point called a
vertex
Each bridge is represented by a curve called an
arc or edge
A
B
C
D
5
Interpret the problem with a GRAPH
Each land mass is represented by a point called a
vertex
Each bridge is represented by a curve called an
arc or edge
A
B
C
D
6
Interpret the problem with a GRAPH
Each land mass is represented by a point called a
vertex
Each bridge is represented by a curve called an
arc or edge
A
B
C
D
7
Interpret the problem with a GRAPH
Each land mass is represented by a point called a
vertex
Each bridge is represented by a curve called an
arc or edge
A
B
C
D
8
Interpret the problem with a GRAPH
Each land mass is represented by a point called a
vertex
Each bridge is represented by a curve called an
arc or edge
A
B
C
D
9
Interpret the problem with a GRAPH
Each land mass is represented by a point called a
vertex
Each bridge is represented by a curve called an
arc or edge
A
A
B
B
C
C
D
D
This is called a NETWORK
10
The question Can you cross each bridge exactly
once? Becomes Can you trace every arc (edge)
exactly once? That is Can you draw this
without lifting your pencil and without
retracing any arc?
A
B
C
D
11
definition A network is said to be TRAVERSABLE
if you can trace each arc exactly once without
lifting your pencil.
12
definition A network is said to be TRAVERSABLE
if you can trace each arc exactly once without
lifting your pencil.
This network is traversable.
The Konigsburg network is not traversable. How
do we know this?
13
Every vertex in a network can be classified as
either even or odd depending on the number of
arc endings that meet in the vertex.
2
3
1
4
14
Every vertex in a network can be classified as
either even or odd depending on the number of
arc endings that meet in the vertex.
2
1
3
5
4
15
Every vertex in a network can be classified as
either even or odd depending on the number of
arc endings that meet in the vertex.
16
Every vertex in a network can be classified as
either even or odd depending on the number of
arc endings that meet in the vertex.
If you start in an odd vertex, you will end
outside of the odd vertex.
end
out
start
in
out
in
out
17
Every vertex in a network can be classified as
either even or odd depending on the number of
arc endings that meet in the vertex.
If you start in an odd vertex, you will end
outside of the odd vertex.
start
and if you start outside an odd vertex, you
will end in the odd vertex.
end
in out in out in
18
Every vertex in a network can be classified as
either even or odd depending on the number of
arc endings that meet in the vertex.
If you start in an odd vertex, you will end
outside of the odd vertex.
start
and if you start outside an odd vertex, you
will end in the odd vertex.
end
A TRAVERSABLE NETWORK has AT MOST 2 ODD
VERTICES. To traverse the network, you must start
in one odd vertex and end in the other.
19
The Konigsburg bridge network is NOT traversable
because there are 4 odd vertices.
A
A
B
B
C
C
D
D
20
If a network has all even vertices, then it is
traversable. To traverse the network, you can
start at any vertex and you will end where you
started.