2. The Konigsberg Problem
Konigsberg was a city in Prussia (now
Kaliningrad, Russia)
This city was situated in both sides of
the Pregel River and included 2 large
islands-which were connected to each
other, or to the mainland portions of
the city, by 7 bridges.
3. The Seven Bridges of Königsberg
The old town of Königsberg has seven
bridges:
4. PROBLEM :
Can you take a walk through
the town, visiting each part of
the town and crossing each
bridge exactly once ?
6. There are 4 land areas of the town-on the
mainland north of the river, on the mainland
south of the river and the 2 islands.
Let us label them A,B,C, & D.
To visit each part of the town you
should visit the points A,B,C & D.
Let us label the bridges p,q,r,s,t,u & v
You should cross each bridge exactly
once.
8. Now we again simplifying this picture
by points and curves(or lines) where
land areas are indicated by points
and bridges are indicated by lines.
9. We call this kind of picture
a graph – the points are
called vertices and the lines
or curves are called edges.
Our goal of finding “a walking
tour that crosses each bridge
once” is now matter of tracing
out all the edges without lifting
our pencil.
10. Euler’s solution to the original
bridge problem
Euler realized that trying to find a path by
drawing the layout of the bridges and
connecting them various ways would take a
lot of time and would not necessarily result in
a path that fulfilled the criteria.
Instead he made the problem into a graph
problem.
In the corresponding graph each vertices are
connected with odd number of edges(such
vertices are called odd vertices).
11. If you start the path at one of the
vertices, you can visit each vertices but
one of the bridges is left out of the path
as in the below example :
12. The path can be started at a different
vertex or travel in a different order
between the vertices, but there will
always be at least one edge not left out.
• This is because if an odd vertex is the
starting point, you have to leave, then come
back, then leave again for another vertex.
• If that other vertex is also an odd vertex,
then you have to leave after coming in and
then come back again and there is no other
edge to allow you to leave again for the other
vertices.
• Thus if there are odd vertices, the path
must start and end at an odd vertex.
13. • So there can be at most 2 odd
vertices.
• This graph has four odd vertices
so it is not possible to have a path
where each edge is used exactly
once.
14. “ THERE IS ABSOLUTELY
NO PATH OVER THE
KONIGSBERG BRIDGES ”
Using this reason, Euler said that
15. This problem and solution or
lack of solution to it became the
start of the famous branch of
Mathematics named
“GRAPH THEORY”
