Eulerian Circuits

A Warm Up Problem• Jenny & John were at a Math Circles event with three other couples.

As people arrived, various handshakes took place.

• No one shook hands with the person they came with, and no one shook their own hand (of course!)

• After all of the introductions had been made, Jenny asked the other seven people how many hands each shook. Surprisingly, they all gave different answers.

• How many hands did John shake? Note: This problem comes from John Grant McLoughlin & Richard Hoshino

Types of Graphs

Bar Graph

Pie Chart

Line Graph

Histogram

Terminology

Graphs can be used to represent relationships between objects. The vertices are the objects and if two vertices are related, we put an edge between them.

• The degree of a vertex is the number of edges coming out of it. The degree of vertex 4 is 3,the degree of vertex 5 is 1.

Eulerian Circuits and Walks• A graph has an Eulerian circuit if you can start

at vertex, v, and get to all the other vertices, using each edge exactly once, ending at vertex v.

• An Eulerian walk is like an Eulerian circuit, but you need not end at the vertex you started at.

Halifax Harbour• Can you start at any island, visit all the other islands,

using each ferry only once, ending at the island you started at?

• No!• Why not?

Halifax Harbour• Can you start at any island, visit all the other islands,

using each ferry only once?You need not end where you started

• No!• Why not?

Why Doesn’t it Work?• For a vertex that is not our start/finish, we

enter that vertex, then immediately leave it. This means that the vertices that are not our start/finish points have........

• EVEN DEGREE!• Since all the vertices are ODD degree, we

always get stuck at an islandbefore we reach our endpoint.

The 7 Bridges of Konigsberg

Leonhard Euler was a Swiss mathematician. He started to go blind in 1735 but managed to produce a mathematical paper a week until 1775!

Problem: Find a walk through the city that would cross each bridge only once. The islands could not be reached by any route other than the bridges, and every bridge must have been crossed completely. Euler posed this problem back in 1735. It was the first graph theory problem! (This is why they are called “Eulerian” circuits and walks)

Pirate Cove

Conclusions about Eulerian circuits and walks

• If a graph has all even vertices, then the graph has an Eulerian circuit.

• If a graph has 2 vertices of odd degree and the rest are even, then the graph have an Eulerian walk which starts at one odd degree vertex and ends at the other odd degree vertex.

You’re a Mathematician!

• Euler came up with the theorem: “A graph, G has an Eulerian Circuit if and only if G has all even degree vertices”

• And you discovered this same result all on your own!!

Which of the following graphs have an Eulerian Circuit?

12

34

Which of the following graphs have an Eulerian Walk?

12

34

DNA

KOLAM

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