3–Planargraphs:afewproblems Remainingtime 2–Planargraphs:ashortintroduction 15mins 1–Thethreehousesandthreeutilitiesproblem 10mins • • Materials: • • Objectives: • Planargraphs Season2Episode21Timeframe2periods

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Season 2 • Episode 21 • Planar graphs

⁰

Planar graphs

Season 2

Episode 21 Time frame 2 periods

Prerequisites :

^Basi^notions^and^voabulary^about^graphs.

Objectives :

•

^Disover ^the ^onept ^of ^planar ^graphs.

•

Ûse ^the ^main ^results âbout ^planar ^graphs^to ^solve â^few ^problems.

Materials :

•

^Lesson.

•

^Problems.

•

^Beamer.

1 – The three houses and three utilities problem ^{10 mins}

The problem of the three houses and three utilities is presented to the students. They

have 10minutes to tryto solve it.

Suppose we have three houses and three utility outlets (eletriity, gas and

water) situatedas shown onthe piture below. Is it possible toonnet eah

utility toeah of the three houses withoutthe lines or mains rossing?

2 – Planar graphs : a short introduction 15 mins

The main notions and results about planar graphs are presented by the teaher with a

beamer.Students are handed out alesson at the end of the presentation

3 – Planar graphs : a few problems Remaining time

Working by pairs or groupsof three, students have tosolvea few problems about planar

graphs.

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Planar graphs ^Episode ²¹

Document Lesson

A planar graph is a graph that an be drawn in the plane in suh a way that no two

edges interset exept at a vertex. A plane graph is a graph that is atually drawn with

no interseting edges.

Plane Planar Non-planar

Let

G

^be â ^planar ^graph. Îts êdges ^dene ^regions ôf ^the ^plane. ^The ^verties ând êdges

that are inidenttoa regionmake up itsboundary.Forexample, the planargraphshown

as anexample denes 3regions of the plane : the outer regionmust always be ounted.

When

G

îs â ^planar ^graph, ^we ûsually ^denote

p

^its ^number ^of ^verties,

q

^its ^number ^of

edges and

r

^its^numbers ^of ^regions.

Theorem 1 Euler’s formula

Let G be a connected plane graph with p vertices, q edges and r regions. Then p − q + r = 2 .

Theorem 2

Let G be a connected plane graph with p vertices, q edges, where p > 3 . Then

q 6 3 p − 6 .

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Planar graphs

Season 2

Episode 21 Document Problems

Problem 1

Show that the graphs of the tetrahedron,the ube and the otahedron are planar.

Problem 2

Drawall possible plane graphswith 1, 2or 3verties.

Problem 3

Drawa non planar graph with8 verties.

Problem 4

Use polydrons to build a dodeahedron and an iosahedron, then draw plane graphs for

these twosolids.

Problem 5

For what values of

n

^is ^the ^omplete ^graph

K _n

^planar^? ^Fôr êah ^value ôf

n

^suh ^that ^it

is so,draw

K _n

^as ^a^symmetri ^plane^graph.

Problem 6

A graphan beobtained fromanygeographi mapby replaingeveryregionbyavertex,

and onneting two verties by an edge exatly when the two regions share a border

segment (not just a orner).

1

^. ^What ^speial ^property ^exhibits ^any ^graph ^built ⁱⁿ^this ^way^?

2

^. ^Restate ⁱⁿ ^the ôntext ôf ^graph ^theory ^the ^famous ^Fôur ôlor ^Theorem ^: ^Given

any separation of aplane into ontiguous regions, alled amap, the regionsan be

olored using at most four olors so that no two adjaent regions have the same

(4)

Problem 7

Use graph theory and planar graphsto re-statethe three houses problem.

Problem 8

Let

K 3 , 3

^be^theômplete^bipartite^graph^shown ^below.^Theâimôf^thisêxeriseîs^to^prove

that

K (3, 3)

^is^not ^planar.

As usual, we all

p

^the ^number ^of ^verties ^of ^the ^graph,

q

îts^number ôf êdges ând

r

^the

number of regions dened by its edges. Moreover, we sum the number of edges lying on

the boundaryof eah region, forall regionsof

K (3, 3)

^,^and ^denote^this ^number

N

^.

1

^. ^Give^the ^value ^of

p

^and

q

^for

K 3 , 3

^.

2

^. ^Suppose ^that

K 3 , 3

îs ^planar. ^Then, ^dedue ^from Êuler's ^formula ^the ^value ôf

r

^for

any plane representation of this graph.

3

^. ^Prove ^that ^no representation of

K 3 , 3

ân ôntain â ^triangle. ^Dedue ^from ^this ^fat

an inequality about

N

^and

r

^.

4

^. ^Explain ^why

N 6 2q = 18

^.

5

^. ^Dedue ^from ^the ^previous ^questions ^an^upper^bound^for

r

^.

6

^. ^Conlude.

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Planar graphs

Season 2

Episode 21 Document Solutions

Problem 1

Toshow thatthe graphsof the tetrahedron, the ube and the otahedron are planar, we

just haveto drawthem planegraphs of these threesolids. These plane graphsare shown

below.

b b b

b b b b b

b b b

b b

Problem 2

There are 2 plane graphswith 1 vertex :

b b

There are 6 plane graphswith 2 verties:

b b b b b b

b b b b b b b b

(6)

There are 20 plane graphswith 3 verties :

b b

b b b

b

b b

b b b

b

b b b b

b b b

b

b b b

b b

b b b

b b

b b b

b

Problem 3

A simple example of non planargraph with 8 vertiesis the omplete graph

K 8

^:

b b b b

b

b b

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Season 2 • Episode 21 • Planar graphs

⁶

Plane graph of a dodeahedron:

b b

b

b b

b b b b b

b b

b

b b b b

b

Plane graph of aniosahedron :

b b b

b

b b b b b

b b

Problem 5

The omplete graph

K _n

^is^planar ^only^for

n 6 4

^. ^These ^graphs ^are ^shown ^below.

b b b b

b b b b b b

Problem 6

1

^. Âny ^graph^builtⁱⁿ^this ^way îs^plane.Îndeed,âs^frontiers ânnot^rossêahôtherôn

a map, edges annotdoit either ina graph built froma map.

2

^. Înâny ^plane^graph,^the ^vertiesân ân^beôloredûsingât^most^fourôlors^so^that

no two adjaent vertieshave the same olor.

Problem 7

Consider a graph made of two groups of three verties, suh that eah vertex of eah

group is adjaent toeah vertex of the other group, but to no vertex of the same group.

This graph is not planar.

Problem 8

1

^. ^F^or

K 3 , 3

^,

p = 6

^and

q = 9

^.

2

^. ^F^or ^any ^plane representation of

K 3 , 3

^, ^we ^would ^still ^have

p = 6

^and

q = 9

^. ^Then,

aording to Euler'sformula,

p − q + r = 2

^, ^so

6 − 9 + r = 2

^,^therefore

r = 5

^.

(8)

3

^. ^First,^we^an ^notie^that

K 3 , 3

îs^madeôf^two^groupsôf^verties,êah ône^with ^three

verties, and suh that eah vertex of eah group is adjaent to eah vertex of the

other group, but to no vertex of the same group. Now, onsider three verties

A

^,

B

^,

C

ⁱⁿ

K 3 , 3

^. Ât ^least ^two ôf ^them ^must ^be ⁱⁿ ^the ^same ^group. Îf ît's êxatly^two,

say

A

^and

B

^, ^then ^these ^two ^verties ^are ^not ^adjaent, ^so

A

^,

B

^and

C

^don't ^make

a triangle.Ifthe three verties are in the same group,then none is adjaent toany

other, so

A

^,

B

^and

C

^don't ^make ^a ^triangle. ^Then, ^no representation of

K 3 , 3

^an

ontain a triangle.

This means that any region in suh a plane graph will be bounded by at least

4

edges, so

N > 4 r

^.

4

^. Âny êdge ân ^bound ât ^most ^two ^regions,^so

N 6 2 q

^.^But

q = 9

^, ^so

N 6 2 q = 18

^.

5

^. ^We^know^that

N > 4r

^and

N 6 2q = 18

^,^so

4r 6 N 6 18

^,

4r 6 18

^,

r 6 3.5

^.

6

^. ^If ^question ² ^we've ^seen ^that ^if

K 3 , 3

^is ^planar, ^then

r

^must ^be ^equal ^to

5

^. ^But

in question

5

^we ^found ^that ⁱⁿ ^any representation of

K 3 , 3

^,

r 6 3.5

^. ^These ^two

inequalitiesontradit eahother, so

K 3 , 3

^is ^not ^planar.

3–Planargraphs:afewproblems Remainingtime 2–Planargraphs:ashortintroduction 15mins 1–Thethreehousesandthreeutilitiesproblem 10mins • • Materials: • • Objectives: • Planargraphs Season2Episode21Timeframe2periods

Season 2 • Episode 21 • Planar graphs

Planar graphs

Season 2

Episode 21 Time frame 2 periods

Prerequisites :

Objectives :

•

•

Materials :

•

•

•

1 – The three houses and three utilities problem 10 mins

2 – Planar graphs : a short introduction 15 mins

3 – Planar graphs : a few problems Remaining time

Planar graphs Episode 21

Document Lesson

G

G

p

q

r

Theorem 1 Euler’s formula

Let G be a connected plane graph with p vertices, q edges and r regions. Then p − q + r = 2 .

Theorem 2

Let G be a connected plane graph with p vertices, q edges, where p > 3 . Then

q 6 3 p − 6 .

Planar graphs

Season 2

Episode 21 Document Problems

n

K n

n

K n

1

2

K 3 , 3

K (3, 3)

p

q

r

K (3, 3)

N

1

p

q

K 3 , 3

2

K 3 , 3

r

3

K 3 , 3

N

r

4

N 6 2q = 18

5

r

6

Planar graphs

Season 2

Episode 21 Document Solutions

b b b

b b b b b

b b b

b b b

b b

b b

b b

b b b b b b

b b b b b b b b

b b

b b b

b b b

b b b

b

b b

b b b

b b b

1 – The three houses and three utilities problem ^{10 mins}

Planar graphs ^Episode ²¹

K _n

K _n

K _n