Koenigsberg bridge problem
It is the Pregel River divided
Koenigsberg into four distinct
sections. Seven bridges
connected the four portions of
Koenigsberg.
It was a popular pastime for the
citizens of Koenigsberg to start in
one section of the city and take a
walk visiting all sections of the city,
trying to cross each bridge exactly
once and to return to the original
starting point.
How did it start?
 In 1735, a Swiss Mathematician
Leonhard Euler became the first person
to work in graph theory by solving the
Koenigsburg bridge problem.
 Discovered a simple way to determine
when a graph can be traced.
Definition
Trace-to begin at some vertex
and draw the entire graph
without lifting your pencil
and without going over any
edge more than once.
On a piece of paper draw these 2 pictures.
Chap 4 sec 1
a)
b)
Exercise 1
Place your pencil on any dot and
trace the figure completely without
lifting your pencil and without
tracing any part of any line twice.
Which of the two can be done?
Solution
Fig. A can be traced.
Fig. B cannot be traced.
Definitions
 Graph- consists of a finite set of points
 Vertices – are points on the graph
 Edges- are lines that join pairs of vertices
 Connected- if it is possible to travel from
any vertex to any other vertex of the graph
by moving along successive edges.
 Bridge- in a connected graph is an edge such
that if it were removed the graph is no
longer connected.
A A
E
C
D
B
F
Connected graph, the vertex CD is a bridge
A
D
B
C
Nonconnected graph
E
Odd and Even Vertex
 Odd – The graph is odd if it is an
endpoint of an odd number of edges of
the graph.
 Even- The graph is even if it is an
endpoint of an even number of edges of
the graph.
Determine which vertices are even
and which are odd.
A
B
C
D
Solution
 Vertex A is odd
 Vertex B is odd
 Vertex C is odd
 Vertex D is odd
Determine which vertices are odd
and even
A
B
C
D
Solution
 Vertex A is odd
 Vertex B is odd
 Vertex C is even
 Vertex D is even
Euler’s Theorem
A graph can be traced
if it is connected and
has zero or two odd
vertices.
Which of the graphs can be traced?
A
B
A
B
C
C
D
D
E
Solution
 Fig. 1 Cannot be traced. (all odd)
 Fig. 2 Can be traced by Euler’s theorem.
Note
 If a graph has 2 odd vertices, the
tracing must begin at one of these
and end at the other.
 If all vertices are even, then the
graph tracing must begin and end at
the same vertex. It does not
matter at which vertex this occurs.
Definitions
 Path- in a graph is a series of
consecutive edges in which no edge is
repeated.
 Euler path- A path containing all the
edges of a graph.
Euler circuit- An Euler path that
begins and ends at the same vertex.
Eulerian graph-A graph with all
even vertices contains an Euler
circuit
Find Euler’s path and Euler’s
circuit for the two fig. below.
A
B
A
C
B
E
D
C
D
F
E
G
H
I
Solution
 Fig. 1 (star)
 Euler’s path - ADBECA
 Euler ‘s circuit - ADBECA
 Fig. 2
 Euler’s path – CABCDEHIDFG
 Euler’s circuit – There is none, because G and C are both
odd vertices, we must begin at one and end at the other.
What is Euler’s circuit used for?
How many of you ride the
pubic transportation?
Efficient routes.
Map Coloring
Eulerizing a Graph
 1. The graph must have all even vertices.
 2. If a graph has an odd vertex, then we
will add some edges to make that vertex
an even vertex.
 3. We want to begin and end at the
same vertex.
 4. We do not want to travel on the same
edge twice.
Find and efficient route.
Thank you