1. Vertex-Edge Graphs Euler Paths Euler Circuits

2. The Seven Bridges of Konigsberg

3. Euler (pronounced “oiler”) Paths Vocabulary Theory Problem and Story Examples and Non-Examples Try Some Puzzles

4. Vocabulary Vertex: point (plural-vertices) Edge: segment or curve connecting the vertices Odd vertex: a vertex with an odd number of edges leading to it Even vertex: a vertex with an even number of edges leading to it Euler path: a continuous path connecting all vertices that passes through every edge exactly once Euler circuit: an Euler path that starts and ends at the same vertex

5. Vertex Point Name the points (vertices) on this vertex-edge graph

6. Edges segment or curve connecting the vertices What are the edges on this vertex-edge graph?

7. Odd Vertex a vertex with an odd number of edges leading to it What are the odd vertices on this vertex-edge graph?

8. Even Vertex a vertex with an even number of vertices leading to it What are the even vertices on this vertex-edge graph?

9. Euler Path a continuous path connecting all vertices that passes through every edge exactly once Is this vertex-edge graph an Euler path? Why, or why not?

10. Euler Circuit an Euler path that starts and ends at the same vertex Is this vertex-edge graph an Euler circuit? Why, or why not?

11. Another Example Is this vertex-edge graph an Euler path? Why or why not? Is this vertex-edge graph an Euler circuit? Why, or why not?

12. Euler’s Theorem, or Rules  If a graph has more than two odd vertices, then it does not have an Euler path. If a graph has two or fewer odd vertices, then it has at least one Euler path. If a graph has any odd vertices, then it cannot have an Euler circuit. If every vertex in a graph is even, then it has at least one Euler circuit.

13. Let’s Take Another Look How many odd vertices in this vertex-edge graph? According to Euler’s Theorem, can this be an Euler path? Can it be an Euler circuit?

14. Let’s Look at the Other One How many odd vertices in this vertex-edge graph? According to Euler’s Theorem, can this be an Euler path? Can it be an Euler circuit?

15. Pencil Drawing Problem -Euler Paths Which of the following pictures can be drawn on paper without ever lifting the pencil and without retracing over any segment?

16. Pencil Drawing Problem -Euler Paths Graph Theoretically: Which of the following graphs has an Euler path? First, identify the points.

17. Pencil Drawing Problem -Euler Paths Answer: the left but not the right. startfinish 12 3 4 5 6

18. Sources Glencoe 2008 Resource https://www.montgomeryschoolsmd.org/c urriculum/math/IM%202008/Unit%203/ Extension%20-%20Vertex- Edge%20Graphs.pdfhttps://www.montgomeryschoolsmd.org/c urriculum/math/IM%202008/Unit%203/ Extension%20-%20Vertex- Edge%20Graphs.pdf