Above and Beyond: Graph TheoryAuthor(s): MA A & AS SubcommitteeSource: Mathematics in School, Vol. 30, No. 3 (May, 2001), pp. 12-14Published by: The Mathematical AssociationStable URL: http://www.jstor.org/stable/30212161 .

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Editor's Note: This is the third article in the series 'Above and Beyond'. The series is aimed at able pupils from years 10 and II who may complete work quickly and welcome more challenging ideas. We anticipate that they will enable pupils to gain deeper insight into the topics and enrich their curriculum provision.

AJE and BEYL

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by MA A & AS Subcommittee

A graph is a diagram which joins points, called vertices, with lines, called edges.

Here are three examples:

A B

D C

A B

D/C

A B

D C Graph 1

Graph 2 Graph 3

The degree of each vertex is defined to be the number of

edges coming out of the vertex.

Looking at the examples above:

* In graph 1, there are 4 edges and 4 vertices. Each vertex has degree 2.

* In graph 2, there are 5 edges and 4 vertices. A and C have degree 2 and B and D have degree 3.

* In graph 3, there are 8 edges and 5 vertices. E has degree 4 and the other vertices have degree 3.

* 1. For each of the graphs below, write down:

(i) the number of edges

(ii) the degree of each vertex

(iii) the total of the degrees.

* 2. Is there a connection between the number of edges of a graph and the total of the degrees of the vertices?

Imagine trying to construct a graph, starting with the vertices. The graph is constructed by making sure every edge is drawn once and only once. Can you always construct a graph continuously (i.e. without taking your pen off the

paper)?

Look back at Graph 3 in the examples at the beginning. Starting with the 5 vertices, you cannot construct the graph continuously without drawing an edge twice. But, you can construct Graphs 1 and 2 continuously.

(a) (b) (c) (d) N ) r

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12 Mathematics in School, May 2001 The MA web site www.m-a.org.uk

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Graph Theory In general, when constructing graphs, there are three main types:

* A graph that can be drawn continuously starting at any vertex and finishing at the starting point is called Eulerian. (Graph 1 is Eulerian.)

* A graph that can be drawn continuously but only starting at certain vertices is called semi-Eulerian. (Graph 2 is semi-Eulerian).

* A graph that cannot be drawn continuously is non- Eulerian.

* 3. What are the types of each of the graphs in Question 1?

Conditions for each type:

* Eulerian - all the vertices have even degrees. * Semi-Eulerian - exactly 2 vertices have odd degrees. * Non-Eulerian - more than 2 vertices have odd degrees.

* 4. Check your answers for Question 3 using these conditions.

* 5. Explain why it is impossible to have a graph with only one odd degree vertex.

* 6. Can you explain these conditions? - try to write a couple of sentences to explain each one.

* 7. The Konigsberg Bridge Problem

a::a!i~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~~~~~a::

..".............

;~. .. . :; ;. i

....

Seven bridges in K6nigsberg are arranged as shown.

Is it possible to plan a walk in which each of the seven bridges is crossed once and only once?

Keywords: Graph Theory; Upper Secondary; Enrichment.

Authors MA A & AS Subcommittee. c/o Charlie Stripp, 26 Thompson Road, Exeter EX1 2UB. e-mail: charlie@mei-distance.com

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Mathematics in School, May 2001 The MA web site www.m-a.org.uk 13

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Graph Theory

Solutions

2. 2 x number of edges = the total of the degrees of the vertices.

3. (a) Eulerian

(b) Non-Eulerian

(c) Non-Eulerian

(d) Semi-Eulerian

(e) Eulerian

5. The total of the degrees of all of the vertices must be even - each edge has a beginning and an end.

6. Eulerian: To be able to draw a graph continuously, starting from any vertex, each time an edge is followed to a vertex, there must be another unused edge available along which to leave the vertex. This means that each vertex must have an even degree.

Semi-Eulerian: To draw a graph continuously, it is possible that you could 'leave' the first vertex one more time than

you 'enter' it and 'enter' the last vertex one more time than you 'leave' it. This means that the start and finish vertices could have an odd degree. Note that this also means that such graphs can only be drawn continuously if you start at one odd degree vertex and finish at the other.

Non-Eulerian: For any graph which does not have either

nought or two vertices with odd degree, there will be at least one vertex which, at some point in drawing the graph, will have no unused edge available to 'leave' along once it has been 'entered'. Thus it will be impossible to draw such a

graph continuously.

7. It is not possible. The bridges and routes between them form a graph equivalent to this:

Each node represents a region of K6nigsberg. Each edge represents a bridge.

This graph has four odd degree vertices and so is non- Eulerian. M

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14 Mathematics in School, May 2001 The MA web site www.m-a.org.uk

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