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Date	November 2019	Marks available	4	Reference code	19N.3.AHL.TZ0.Hdm_4
Level	Additional Higher Level	Paper	Paper 3	Time zone	Time zone 0
Command term	Find	Question number	Hdm_4	Adapted from	N/A

Question

$G$ is a simple, connected graph with eight vertices.

$H$ is a connected, planar graph, with $v$ vertices, $e$ edges and $f$ faces. Every face in $H$ is bounded by exactly $k$ edges.

Write down the minimum number of edges in $G$ .

[1]

a.i.

Find the maximum number of edges in $G$ .

[2]

a.ii.

Find the maximum number of edges in $G$ , given that $G$ contains an Eulerian circuit.

[2]

a.iii.

Explain why $2e = kf$ .

[2]

b.i.

Find the value of $f$ when $v = 9$ and $k = 3$ .

[3]

b.ii.

Find the possible values of $f$ when $v = 13$ .

[4]

b.iii.

Markscheme

7 (a tree) A1

[1 mark]

a.i.

$\frac{{8 \times 7}}{2}\left( {7 + 6 + 5 + 4 + 3 + 2 + 1} \right)$ (a complete graph) (M1)

$= 28$ A1

[2 marks]

a.ii.

$\frac{{8 \times 6}}{2}$ (since every vertex must be of degree 6) (M1)

$= 24$ A1

[2 marks]

a.iii.

counting the edges around every face gives $k f$ edges A1

but as every edge is counted in 2 faces R1

$\Rightarrow kf = 2e$ AG

[2 marks]

b.i.

using $v - e + f = 2$ with $v = 9$ M1

EITHER

substituting $2e = 3f$ into $2\left( 9 \right) - 2e + 2f = 4$ (M1)

substituting $e = \frac{{3f}}{2}$ into $9 - e + f = 2$ (M1)

THEN

$18 - f = 4$

$f = 14$ A1

[3 marks]

b.ii.

$2v - kf + 2f = 4$ (or equivalent) M1

when $v = 13$

$\left( {k - 2} \right)f = 22$ or $\left( {2 - k} \right)f = - 22$ A1

EITHER

$\left( {k - 2} \right)f = 1 \times 2 \times 11$ M1

substituting at least two of $k = 13{\text{, }}4{\text{, }}3$ into $f = \frac{{22}}{{k - 2}}$ (or equivalent) M1

THEN

$f = 2{\text{, }}11{\text{, 22}}$ (since $f > 1$ ) A1

[4 marks]

b.iii.

Examiners report

[N/A]

a.i.

[N/A]

a.ii.

[N/A]

a.iii.

[N/A]

b.i.

[N/A]

b.ii.

[N/A]

b.iii.

Syllabus sections

Topic 3—Geometry and trigonometry » AHL 3.14—Graph theory

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Find the maximum number of edges in $G$ , given that $G$ contains an Eulerian circuit.
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Write down the minimum number of edges in $G$ .
19N.3.AHL.TZ0.Hdm_4b.ii:
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Topic 3—Geometry and trigonometry

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