• Königsberg was a city in Prussia
situated on the Pregel River (Today,
the city is named Kaliningrad, and
is a major industrial and
commercial center of western
• A river Pregel flows around the
island Keniphof and then divides
• Seven bridges spanned the various
branches of the river, as shown.
• It became a tradition to try to walk
around the town in a way that only
crossed each bridge once, but it
proved to be a difficult problem.
Determine whether it’s possible to walk across all the bridges exactly once in
returning back to the starting land area.
•In 1736 Euler proved that the walk was not possible to do. No Eulerian walk of
the Konigsberg bridge problem since all four vertices are of odd edges.
• He proved this by inventing a kind of diagram called a network, that is made up
of vertices (dots where lines meet) and arcs (lines). Define the degree of a vertex
to be the number of edges incident to it.
•He used four dots (vertices) for the two riverbanks and the two islands. These
have been marked A, B and C, D. Euler showed that there is a walk starting at any
vertex, going through each edge exactly once and terminating at the start vertex
iff the degree of each vertex is even. This walk is called Eulerian.
•The seven lines (arcs) are the seven bridges.
3 bridges (arcs) join to riverbank A, and
3 join to riverbank B. 5 bridges (arcs)
join to island C, and 3 join to island D.
All the vertices have an odd number of
arcs, so they are called odd vertices. (An
even vertex would have to have an even
number of arcs joining to it).
WHY IS IT IMPOSSIBLE!
Euler's solution involved the observation that when a vertex is "visited" in the
middle of the process of tracing a graph, there must be an edge coming into the
vertex, and another edge leaving it and so the order of the vertex must be an even
This must be true for all but at most two of the vertices--the one you start at, and
the one you end at and so a connected graph is traversable if and only if it has at
most two vertices of odd order.
Now the NETWORK graph shows that there are more than two vertices of odd
order and therefore the graph cannot be traced.
It is concluded that the desired walking tour of Königsberg is impossible.
ANY IDEAS FOR THE CRISIS…?
Suppose the citizens of Königsberg decided
to build an eighth bridge, as in the diagram
shown than it would be possible walking
without traversing any bridge twice.(ONLY
TWO VERTEX ARE ODD ORDER.)
Adding a ninth bridge to the diagram
above will make the walking tour once
A graph, G, consists of two sets, V
V is a finite, nonempty set of
E is set of pairs of vertices called
The vertices of a graph G can be
represented as V(G).
Likewise, the edges of a graph, G,
can be represented as E(G).
Graphs can be either undirected
graphs or directed graphs.
For a undirected graph, a pair of
vertices (u, v) or (v, u) represent the
For a directed graph, a directed pair
<u, v> has u as the tail and the v as
the head. Therefore, <u, v> and <v,
u> represent different edges.
The degree of vertex = No. of edges connected with a vertex
Simple path = Such a way that each vertex is traversed once.
Euler Path = Such a way that each edge is traversed once.
From above graphs it is concluded that Euler path is possible when odd degree
vertex are either 0 or 2.
In case of Konnisberges Bridge – all the four vertices were odd degree and
hence as per above conclusion – Euler path is not possible.
By adding one more bridge - two vertices are even degree and two are odd
degree- Euler path is possible
APPLICATION OF GRAPH THEORY
GPS (Global Positioning System ) uses graph to find shortest path.
Airlines uses closed paths and look for alternatives for cancelled flights.
Power grids use it for transfer of electricity without overloading lines.
Graphs are used to design computer networks.
Analysis of electrical circuits
Finding shortest routes
Identification of chemical compounds
Cybernetics: a branch where human beings use various types of networks:
electric or mechanical
Social Sciences, and so on …
SUMMARY OF TRAVERSABLE GRAPH
Number Of Odd
Start anywhere and end
back at the start vertex.
Start at one odd and end at
other odd vertex.
More than 2
A graph is a collection of vertices and edges.
The order of a vertex is the number of the
edges meet at that vertex.
A graph is traversable if and only if it has 0 or 2