DEPARTMENT OF MATHEMATICS
Problem Sheet 1
1. (a) Draw a graph with vertices
v
1
, v
2
, v
3
and
v
4
and edges (
v
1
, v
2
)
,
(
v
1
, v
3
)
,
(
v
2
, v
3
)
,
(
v
2
, v
4
)
and (
v
4
, v
1
).
(b) Write down the number of edges, the degree of each vertex and the sum of the degrees
of the vertices in the graph drawn.
2. The
intersection graph
of a collection of sets
A
1
, A
2
, . . ., A
n
is the graph that has a
vertex for each of these sets and has an edge connecting a pair of vertices if and only if
the sets represented by these vertices have a non-empty intersection.
(a) Construct the intersection graph of the following collection of sets:
A
1
=
{
0
,
2
,
4
,
6
,
8
}
,
A
2
=
{
0
,
1
,
2
,
3
,
4
}
, A
3
=
{
1
,
3
,
5
,
7
,
9
}
, A
4
=
{
5
,
6
,
7
,
8
,
9
}
,
A
5
=
{
0
,
1
,
8
,
9
}
.
(b) Write down the number of edges in the graph constructed in (a).
(c) Write down the degree of each vertex and the sum of the degrees of the vertices in
the graph constructed in (a).
3. Seven towns