Specialist Mathematics Unit 1
Sample learning activity – graph theory constructions and
proofs
Introduction
The module on the Australian Mathematical Sciences page provides an excellent introduction to
the beginning student of graph theory:
http://amsi.org.au/ESA_Senior_Years/SeniorTopic7/7_md/SeniorTopic7a.html
The following collections of related results can be proved after the initial exploration of ideas using
a collection of sample graphs constructed by hand and/or using technology, for example:
http://illuminations.nctm.org/Activity.aspx?id=3550 or http://visualgo.net/mst.html
Part 1
Prove that the sum of the degrees of the vertices of any finite graph is even.
a.
b.
Show that every simple graph has two vertices of the same degree.
Show that if n people attend a party and some shake hands with others (but not with
themselves) then at the end, there are at least two people who have shaken hands with the
same number of people.
n(n  1)
edges.
2
c.
Prove that a complete graph with n vertices contains
d.
Prove that a finite graph is bipartite if and only if it contains no cycles of odd length.
Part 2
a.
Show that any graph where the degree of every vertex is even has an Eulerian cycle. Show
that if there are exactly two vertices of odd degree, there is an Eulerian path from one to the
other.
b. Show that if there are more than two vertices of odd degree, it is impossible to construct an
Eulerian path.
c. Show that in a directed graph where every vertex has the same number of incoming as
outgoing paths there exists an Eulerian path for the graph.
Areas of study
The following content from the areas of study is addressed through this learning activity.
Area of study
Topic(s)
Content dot point
Discrete mathematics
Graph theory
1, 2, 3, 4, 5, 6
© VCAA
graph theory proofs
Outcomes
The following outcomes, key knowledge and key skills are addressed through this task.
Outcome
Key knowledge dot point
Key skill dot point
1
1, 2, 3, 4
1, 2, 3, 4
2
2, 3, 4
2, 3, 4
3
1, 3
1, 4, 5, 9, 10
© VCAA
Page 2