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