1. Konigsberg’s Seven Bridges

2. The Königsberg Bridge Problem The following figure shows the rivers and bridges of Königsberg. Residents of the city occupied themselves by trying to find a walking path through the city that began and ended at the same place and crossed every bridge exactly once.

6. In mathematics, and more specifically in graph theory, a graph is a structure amounting to a set of objects in which some pairs of objects are in some sense “related”. The objects correspond to mathematical abstractions called vertices and each of the related pairs of vertices is called an edge. Typically, a graph is depicted as a set of dots for the vertices, joined by lines for the edges.

7. Graph – is a structure amounting to a set of objects in which some pairs of objects are in some sense “related”. Vertex (vertices) –the points on a graph. Edge (edges) – the lines or curves on a graph. Degree of a vertex – is the number of edges that meet at the vertex.

8. Conjecture – is an educated guess, based on information gathered to date that might be true or false. If the conjecture can be shown true, it becomes a theorem.

9. Euler Circuit – is a circuit that uses every edge of a graph exactly once while starting and ending at the same vertex. Euler path – is a path that uses every edge of a graph exactly once while starting and ending at different vertices.

10. Euler Circuit – is a circuit that uses every edge of a graph exactly once while starting and ending at the same vertex. A graph is an euler circuit if all vertices have an even degree.

11. Euler path – is a path that uses every edge of a graph exactly once while starting and ending at different vertices. An euler path can have only two vertices with odd degrees. For the path to work you must start at one odd degreed vertex and end at the other odd degreed vertex.

12. Implication (for a connected graph)
# of ODD Vertices
Implication (for a connected graph)
There is at least  one Euler Circuit. 1
THIS IS IMPOSSIBLE! (Try it yourself.) 2
There is no Euler Circuit but at least 1 Euler Path.
more than 2
There are no Euler Circuits or Euler Paths.

13. Download this App One touch drawing