1. Warm Up – 3/19 - Wednesday Give the vertex set. Give the edge set.
Give the degree of each vertex.
Is there an Euler path or circuit?

2. Edges and Degrees
The number of edges is exactly half of the sum of all the degrees.
A complete graph is the graph where every vertex has an edge connecting it to every other vertex.

6. Is there an Euler Path or Circuit?
Find the degree of each vertex.
If an Euler circuit exists, give the
sequence of vertices.

7. Solution deg 𝐴 =2 deg 𝐵 =2 deg 𝐶 =2 deg 𝐷 =2 deg 𝐸 =4
There are no odd degrees. Therefore there is an Euler circuit! We can get it starting at A and following: A, B, E, C, D, E, A.

8. Euler Path or Circuit? Find the degree of each vertex.
If an Euler circuit exists, give the
sequence of vertices.

9. Solution deg 𝐴 =3 deg 𝐵 =4 deg 𝐶 =2 deg 𝐷 =4 deg 𝐸 =3 deg 𝐹 =4
There are two odd degrees and the rest are even. This means we have a path! The path must start at an odd degree, we will choose A.
Path: A, B, C, D, B, F, D, E, F, A, E

10. Euler Circuit WS

12. Euler vs. Hamilton
Euler Circuits use all of the edges once and only once and end on the same vertex.
Hamilton Circuits use all of the vertices and end on the same vertex.

13. Hamilton Circuits
Give the Hamilton Circuits that begin with A.

14. Solution
Notice that our two solutions are the reverse of one another!

15. Hamilton Circuits
Give a Hamilton Circuit if it exists. If not, explain why one does not exist.

16. Solution
One does not exist because we would have to cross points B and C twice.
We call the edge BC a bridge. A bridge is an edge
that if removed creates a disconnected graph.