Faculty of Mathematics Centre for Education in

Waterloo, Ontario N2L 3G1 Mathematics and Computing

Grade 7/8 Math CirclesMarch 4/5, 2014

Graph Theory I- Solutions“*” indicates challenge question

1. Trace the following walks on the graph below. For each one, state whether it is a path?

How do you know? (b) and (d) are paths since they do not repeat vertices. Notice

that A is repeated in (a) and (c), and E is repeated in (e).

(a) L-C-E-A-B-A-D

AB

C

DE

FG

HI

J

K

L

(b) H-F-G-J-D-A

AB

C

DE

FG

HI

J

K

L

(c) F-D-A-B-K-E-A

AB

C

DE

FG

HI

J

K

L

(d) F-G-J-H

AB

C

DE

FG

HI

J

K

L

(e) D-A-E-B-K-E-C

AB

C

DE

FG

HI

J

K

L

1

2. Find 10 walks from 1 to 4 in the following graph. How many of these walks are paths?

How many possible paths are there between 1 and 4. (Hint: You may want to redraw

the graph)

1

2

3

4

56

7

8

910

11

There are many walks possible. Students should strive to find at least 10. Remember

that walks can include repeated vertices. There are 8 paths from 1 to 4: 1-4, 1-2-4,

1-2-5-8-3-10-4, 1-2-10-4, 1-6-8-3-10-4, 1-6-8-3-10-2-4, 1-6-8-5-2-4, 1-6-8-5-2-10-4.

3. Find all the cycles in the following graph. There are 6 cycles: 4-8-5-1-4, 1-2-6-5-1,

5-6-9-5, 1-2-6-9-5-1, 4-1-2-6-5-8-4, 1-2-6-9-5-8-4-1.

1 2

4 5 6

89

4. Which edges should be added to the graph below to make it K5? Write the names of

the edges and then draw them on the graph. The edges {a,b}, {a,d}, and {e,b} should

be added so the graph looks like the one below.

a

b

cd

e

2

5. A k-regular graph is a graph with k edges incident

to each vertex. An example of this is the Petersen

Graph which is a type of 3-regular graph.

Luc, Ryan, Vince, Emily, and Nadine go to a party.

When they get there they want to shake hands but

they only have time to shake two other people’s

hands. Draw two different graphs to show how this

can happen. How many handshakes are there? Do

the rules for drawing a graph make sense in this sit-

uation?

a

b

cd

e

f

g

h

i

j

Petersen GraphGraphs will vary. Each vertex should be incident to 2 edges. There should be 5 edges

total. There are 5 handshakes. Yes, the rules make sense since you can not shake hands

with yourself and it does not make sense to have two different handshakes between the

same people since they have already shaken hands once.

6. Sarah has a letter that she wants to give to Emily. But they won’t see each other

because they are in different cities. Answer the following questions. The quickest way

is the following path: Sarah-Vince-Ryan-Luc-Nadine-Emily

Emily

Sarah

Tim

Ryan

Luc

Vince

Nadine

Ishi

Kamil

Lawren

Cass Dalton

(a) Which people must see each other in order for Sarah to get the letter to Emily?

In other words, if these people don’t meet then it is impossible for Sarah to get

the letter to Emily. Ryan and Vince, Luc and Nadine, and Nadine and Emily

must meet.

(b) What do these vertices have in common? They are all bridges.

(c) If Ryan and Vince don’t meet, what changes about the graph (other than removing

an edge)? The graph is not connected. There are now 2 components.

(d) Vince lives far from Sarah so she doesn’t want to give him the letter. What is the

fastest way now? Sarah-Tim-Lawren-Dalton-Cass-Vince-Ryan-Luc-Nadine-Emily

(e) Dalton has a letter that he wants to give to Ishi. What is the fastest way for him

to do this? Dalton-Cass-Vince-Ryan-Kamil or Luc-Ishi

3

7. Are the following graphs planar? If so, draw a planar representation. If not, explain

why. Hint: Only one of these are planar. Try finding K5 or K3,3 subdivisions.

(a) Planar, drawing may vary

1

2

3

4

5

67 8

9

10

11

12

(b) Non-planar: K5 subdivision

1

2

34

56

789

(c) Non-planar: K3,3 subdivision

abc d

ef

gh

(d) Non-planar: K5 subdivisiona

b

c

d

e

f

g

h

8. Redraw each graph to show it is planar.

(a) Drawing may vary.

a b c

d e f

g

h i

(b) Drawing may vary.

12

3

4

5

6

7

8

9

4

9. A word graph has words as the vertices. Two words are adjacent if they differ by 1

letter. Create a connected word graph that contains the following words: log, top, eat,

mud and rag. Answers may vary.

mud.rag.

eat.

top. log.

mug. rug.

bug.

bog.

tap.

tag.

tug.pat.

pot.

pop.

10. Below is a map of South America. Create a graph with the vertices representing coun-

tries and the edges joining those countries which share a land border. So Chile and

Peru are adjacent, but Peru and Uruguay are not. Use your graph to answer the ques-

tions below.

Courtesy of abcteach.com

French GuianaSuriname

GuyanaVenezuela

Columbia

Ecuador

Peru

Bolivia

ParaguayChile

Argentina

Uruguay

Brazil

(a) Find the minimum colouring of the graph, and determine a colour for each country.

Watch out for countries that that share a very small border. Since borders do

not cross, there must be a way to draw a planar graph of the map. The 4-colour

theorem says that any planar graph is 4-colourable. Although it is possible that

a planar graph is less than 4-colourable, there is a subgraph (meaning a smaller

graph within the original) of K4 (consisting of Brazil, Bolivia, Paraguay, and

Argentina). Remember that K4 denotes the complete graph with 4 vertices that

are all adjacent to each other, which requires at least 4 colours.

5

(b) When trading goods over land, a $100 tax is paid to each country which the goods

travel through. So if Columbia sells coffee to Venezuela, the least amount of tax

paid is $100.

i. What is the least amount of tax paid on wool shipped from Ecuador to

Paraguay? $300

ii. What country can trade with the most others for exactly $100? Brazil

iii. Brazil raises it’s taxes to $400. What is the cheapest amount of taxes paid

to ship Venezuelan oil to Paraguay? What about French Guiana to Bolivia?

Venezuela to Paraguay: $400; French Guiana to Bolivia: $500

iv. To encourage trade, Peru will not tax any goods going though the country.

Chile and Suriname want to trade lumber and wheat. What route will result

in the least amount of taxes? How much will they pay? Suriname, Guyana,

Venezuela, Columbia, Peru, Chile; $400

(c) Kevin wants to visit every country in South America on his vacation. He doesn’t

want to visit any country more than once since crossing the border can take a

long time! He will fly into Lima, Peru to begin his trip and fly out of Montevideo,

Uruguay. In what order should he visit each country? Peru, Ecuador, Columbia,

Venezuela, Guyana, Suriname, French Guiana, Brazil, Paraguay, Bolivia, Chile,

Argentina, Uruguay

11. *Two graphs are isomorphic if they are the same graph drawn in a different way. This

means that all the connections are the same, and there are exactly the same number of

vertices, but the vertices may have different names and the graphs may look different.

Determine if the following pairs of graphs are isomorphic. If so, show which vertices

match, or explain why not.

(a) Not isomorphic: there is a cycle of length 3 in the first, but not in the second.

1

2

3

4

5

6

a

b

c

d

e

f

(b) Isomorphic as shown (solutions may vary)

6

1 2 3 4

5 6 7 8

e a b f

g c d h

(c) Isomorphic as shown (solutions may vary)

1

2

34

5 6

7

89

10

a

b

ch

i j

d

ef

g

12. Euler’s Oil is a gas station chain in the country of Mathisfun. The CEO of the company,

Leonhard, wants to visit all his gas stations in the country to ensure that each one is

stocked with enough fuel. However, he doesn’t want to visit any town more than once

as he is a very busy man. Suggest a route Leonhard could take from the head office in

the capital Mathtopia, through each town exactly once, and return to the capital. Note

that despite the intersection of three highways in the centre of the country, there is no

way to change from one road to another (i.e. there is no direct road from Mathtopia

to Gausston, etc.) Answers may vary. One solution is Mathtopia, Circle City, Fraction

Falls, Radius River, Times Town, Equationville, Gausston, Vertex Village, Mathtopia

Mathtopia

Radius River

Gausston

Fraction Falls

Circle City

Equationville

Times Town

Vertex Village

7