a Explain why, if the graph is connected, it must be Eulerian. b Show that the graph need not be connected. c How many edges does the graph have? d What is the greatest value of x for which it is possible to draw a connected graph with the degree sequence 2, 2, 4, x with no loops. A graph has adjacency matrix:
A B C D E F G H
A
B
0
1
0
0
0
1
0
1
⎜1 ⎜ ⎜0 ⎜ ⎜ ⎜0 ⎜ ⎜0 ⎜ ⎜ ⎜1 ⎜ ⎜0
0
1
0
0
0
1
0
1
0
0
0
1
0
1
0
1
0
1
0
1
0
0
0
1
0
1
0
1
0
1
0
0 ⎟ ⎟ 1 ⎟ ⎟ ⎟ 0 ⎟ ⎟ 0 ⎟ ⎟ ⎟ 0 ⎟ ⎟ 1 ⎟
1
0
0
0
1
0
⎛
⎝
1
1 0 0 0 1 0
C
D
E
F
G
H ⎞
PL E
11
⎠
a Write a Hamiltonian cycle for this graph.
b Use Kuratowski’s theorem to show that the graph is non-planar.
M
A connected graph is semi-Eulerian if exactly two of its vertices are of odd degree.
a A graph is drawn with 4 vertices and 7 edges. What is the sum of the degrees of the vertices? b Draw a simple semi-Eulerian graph with exactly 5 vertices and 5 edges, in which exactly one of the vertices has degree 4.
SA
12
c Draw a simple semi-Eulerian graph with exactly 5 vertices that is also a tree. d A simple graph has 6 vertices. The graph has two vertices of degree 5. Explain why the graph can have no vertex of degree 1.
Original material © Cambridge University Press 2018
[© AQA 2016]