Warm Up – 3.20 - Thursday

Quiz Tomorrow

• State if a graph has Euler Circuits or Paths by using the degrees.

• List a sequence of vertices that will give you an Euler Circuit or Path.

• List a sequence of vertices that will give you a Hamilton Circuit or Path.

• State the number of edges given a way to get the degree sum (or vice versa).

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.

• denotesa complete graphwith N vertices.

In a complete graph, if we find the Hamilton Circuits from one starting point, we have found all of them.

Finding all Hamilton Circuits

The number of Hamilton Circuits in a complete graph, is

For

𝐾 5

Notice: When you have found one, with its reverse, you have found 2!

A) Find the number of edges on B) Find the total number of Hamilton Circuits.

Zoomed In

Solution

A) The number of edges is half the degree sum.

• Edges: = 300B) The number of H.C. is equal to This is approximately

SUPERHERO

• The world’s fastest super computer can crank out a list of one quadrillion (million billion) Hamilton circuits per second.

• SUPERHERO can solve in a matter of nanoseconds.

• It takes SUPERHERO 20 years to generate a list for .

• Adding one vertice

Complete Graph Classwork

Mr. Erby is the only principal at school today. He has to visit the High school (B), the Middle school (A), and the three elementary buildings.

The time it takes in minutes to walk between each building is given. What is the quickest amount of time he can get to each building?