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D1 Graph Theory - lesson plan

Lesson objective. To develop students understanding of graph theory

Background. Students will have been introduced to the vocabulary of graphs and will knowthe relevant terminology: vertices, edges, degree (or order) of a vertex, cycle.

By the end of the lesson students will understand the meaning of traversibility how to identify an Eulerian and a semi-Eulerian graph

how to draw a graph given the number of vertices and edges.

Textbook reference Chapter 2, P43 P50 (part)

part (i) Drawing graphs

The table shows the number of vertices of degree 1, 2, 3 and 4 for five different graphs. Raw an example foeach of these graphs

This type of question comes up on

exam papers. A good way to gothrough this is to get students tocome out and draw their graph onthe board. In many cases, therewill be more than one correctdrawing, which is a gooddiscussion point.

Order of vertex 1 2 3 4

Graph 1 3 0 1 0

Graph 2 0 0 4 1

Graph 3 4 0 0 1

Graph 4 0 2 2 2

Graph 5 1 1 3 1

Part (ii) traversable graphs

Which of the following diagrams can be drawn without taking your pencil off the paper?

a b c

How can you predict whether it will be possible to draw a shape without taking your pen off the paper?Where do you start and end each time?What features of the diagram are significant?

Answer

a can be drawn without taking your pen off the paper. You start at any vertex and end at the same vertex.

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b cannot be drawn without taking your pen off the paper

c can be drawn without taking your pen off the paper only if you start at an odd vertex. You will then end ata different odd vertex.

In order for a graph to be TRAVERSABLE, it must have no more than two odd vertices

Part (iii) Konigsberg bridges

The town of Konigsberg in East Prussia was built on thebanks of the river Pregel, with islands that were linkedtogether by seven bridges. The citizens of Konigsbergtried to find a route which would cross each bridge onlyonce and allow them to end their walk where they hadstarted. Can you find a suitable route?

Historical note: Konigsbergis now in Russia and wasrenamed Kaliningrad . Due

to bombing in the secondworld war, only four of thebridges remain.

It is said that the citizens triedunsuccessfully to solve this problemfor many years, but it was not untilLeonard Euler tackled the problemthat it was proved to be impossible.

Solution

The map can be redrawn as a graph withfour vertices (representing the land) andseven edges (representing the bridges). It

can clearly be seen that this gives fourvertices with odd order, so the graph is nottraversable.

Euler published his proof, entitled

Solution Problematis ad geometriamsitus a pertinentis(The solution ofproblem relating to the geometry ofposition), in 1736 and laid thefoundations for Graph Theory.

The problem has become one of finding a cycle which passes along every edge of the graph, called anEULERIAN cycle. Graphs which are traversable are called EULERIAN if all the vertices are even and SEMI-EULERIAN if they contain two odd vertices.

Part (iv) What is the significance of odd vertices?

Find the number of edges and the sum of the degrees of all the vertices of the graphs in part (i)

a 9 edges. 4+4+4+2+2+2 = 18b 9 edges 2+4+3+3+3+3= 18c 9 edges 4+4+3+3+2+2 = 18

Draw 3 connected graphs, each with a different number of vertices and edges.Record the degrees of all the vertices, what do you notice?

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There are always an even number of odd vertices.

From this we deduce that the sum of the degrees of the vertices in a connected graph is always even and isequal to twice the number of edges.

Proof: Each edge has two ends, so it is counted twice in the sum of the degrees, thus the total of thedegrees of the vertices is always even.Because each edge is counted twice, the sum of the degrees of the vertices is always twice the number of

edges.

(extension) This is known as the Handshaking Theorem and can be written deg v = 2e