What does f include in Euler's formula?

All faces, including the outer (unbounded) face.

What is a planar graph?

A graph that can be drawn in the plane with no edges crossing.

How many faces does a tree have?

One, the outer face; for a tree e=v-1.

Effect on f of adding a new vertex joined by one edge?

None; v and e both increase by 1.

Effect on f of subdividing an edge?

None; one vertex and one edge are added.

What is the sum of the face degrees?

2e, because each edge borders two faces (or the same face twice).

Why does a simple planar graph have 2e\ge3f?

Every face has at least 3 edges, as there are no loops or multiple edges.

Why must the graph be connected?

Euler's formula v-e+f=2 applies to connected graphs; disconnected graphs have a different constant.

Euler's formula for planar graphsAQA A-Level Further Maths: Flashcards

Card 1 of 130 of 13 known

Question

State Euler's formula for a connected planar graph.

Tap or press Space to reveal

Tap card or press Space to flip

See all 13 cards

State Euler's formula for a connected planar graph.: $v-e+f=2$
What does $f$ include in Euler's formula?: All faces, including the outer (unbounded) face.
What is a planar graph?: A graph that can be drawn in the plane with no edges crossing.
A connected planar graph has $v=8$ , $e=12$ . Find $f$ .: $f=2-8+12=6$
How many faces does a tree have?: One, the outer face; for a tree $e=v-1$ .
Effect on $f$ of adding a new vertex joined by one edge?: None; $v$ and $e$ both increase by $1$ .
Effect on $f$ of adding an edge between two existing vertices (no crossings)?: $f$ increases by $1$ .
Effect on $f$ of subdividing an edge?: None; one vertex and one edge are added.
What is the sum of the face degrees?: $2e$ , because each edge borders two faces (or the same face twice).
If every face has degree $3$ , which equation links $e$ and $f$ ?: $3f=2e$
Why does a simple planar graph have $2e\ge3f$ ?: Every face has at least $3$ edges, as there are no loops or multiple edges.
Rearrange Euler's formula for $e$ .: $e=v+f-2$
Why must the graph be connected?: Euler's formula $v-e+f=2$ applies to connected graphs; disconnected graphs have a different constant.

Exam questions on Euler's formula for planar graphs

A connected planar graph $G$ has $8$ vertices and $12$ edges.
Every face of $G$ , including the outer face, is bounded by the same number of edges. Find this number.2 marks
A connected simple planar graph has $7$ vertices, and every face, including the outer face, is bounded by exactly $3$ edges.
Find the number of faces, and state how many of them are bounded faces, not counting the outer face.2 marks
A connected planar graph has $9$ vertices, and its edges divide the plane into $8$ faces, including the outer face.
Find the number of edges.3 marks

See the full worksheet

Written by the Exaim team, led by Shaun Daswani (Head of Upper Secondary, Improve ME Institute; MSc Financial Mathematics, Imperial College London; BSc, UCL) and Jason Daswani (operational lead, Improve ME Institute; LSE).