The seven bridge problem of Konigsberg

    The problem appear in 18th century in Konigsberg (Now  known as Kaliningrad and part of Russia).

Konigsberg is a town which is on the banks of Preger river. In here there is a place that around is water like Island. There are 7 bridges for connecting other dry places. One day Konigsberg people thought about one problem. Their goal being to devise in which they could walk around the city, crossing each of the seven bridges only once. Problem is; "Can we find a way that could walk around the city with come back the starting point crossing each of the seven bridges only once?" So they couldn't find it and asked Euler for solving this problem. Euler didn't want to think about problem. He thought that it is for wasting time but then he started to thinking.

In addition in one letter that for his friend he wrote;
   "This question is so banal but seem to me worthy of attention in that neither geometry nor algebra or even the art of counting was sufficient  to solve it."
   So he gave the solution; we can't find a way for our journey.
   
   Euler Solution
   1) He sketched the graph
   2) And called the dry places a, b, c, d. And lines are the small bridges.                 
   3) So he said if we want to get success lines (bridges)  should be even. Because if we should come back to the point that we start, number of lines have to be double.
   In our problem;
   A has  3 lines

   B has  5 lines

   C has  3 lines

   D has  3 lines 

   Therefore  Leonard Euler found;
The bridge system can be different forms but bridges should be double or it can be only max 2 points have odd lines(bridges). According to this solution Euler find the Graph theory.
   
   Graph Theory

   In mathematics, graph theory is the study of graphs, which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of vertices, nodes or points which are connected by edges, arcs, or lines and graph may be undirected, meaning that there is no distinction between the two vertices associated with each edge or its edges may be directed from one vertex to another. According to graph theory the system  must comply this formula;
                                                             V+R-L=1
(V-vertex, R-region, L-line)

   Euler Path and Circuits
   An Euler path, in a graph- is a walk through the graph which uses every edge exactly once.
   An Euler circuit- is an Euler path which starts and stops at the same vertex.

In this system there are 6 Euler path;

A-have 4 lines  (4-1)!
B-2                   (2-1)!
C-2                   (2-1)!     
D-2                   (2-1)!
E-2                   (2-1)!
So,  (4-1)!(2-1)!(2-1)!(2-1)!(2-1)!= 3!1!1!1!1!=6
 
Application of Graph Theory
  We can give some examples for application of graph theory.Chemical models, 
  Google works system, Social networks, 4 color world map

Resources;   https://www.maa.org/press/periodicals/convergence/leonard-eulers-solution-to-the-                                    konigsberg-bridge-problem

                       https://www.mathsisfun.com/activity/seven-bridges-konigsberg.html

                       https://en.wikipedia.org/wiki/Graph_theory
                       http://discretetext.oscarlevin.com/dmoi/sec_paths.html