AMS 301.2 (Spring, 2010)
Estie Arkin
Exam 1 – Solutions
Mean 77.97, median 82.5, top quartile 91, high 100 (2 of them), low 34.
1. (13 points) Are the two graphs shown below isomorphic? If so, give the isomorphism; if not,
give careful reasons for your answer.
1
2
3
4
5
6
7
8
A
B
C
D
E
F
G
H
No. The graph on the left is bipartite (nodes 1,3,5,7 on one side and nodes 2,4,6,8 on the other)
but the graph on the right is not bipartite, it has an odd circuit A,E,G,C,B,A (5 nodes). Also, the
graph on the left is planar, the graph on the right is not.
2. (12 points) Compute the chromatic number (vertex colouring number) of the graph
G
shown
below. Justify your answer! (Show a colouring with
χ
(
G
) colors (label each node with its colour),
and argue that fewer colours cannot suffice.)
A
B
C
D
H
I
J
K
L
M
There are several
K
4
s such as nodes A,C,D,H so at least 4 colours are needed. 4 colours are also
enough (the graph is planar) for example colour 1 are nodes B,C,K; colour 2 nodes A,I,M; colour
3 nodes H,L; colour 4 nodes D,J.
3. (25 points) True or False? If true, give a short proof. If false, give a counterexample:
(a). Every subgraph of a bipartite graph is also bipartite. True. The nodes of the graph are from
two sets “left” and “right”, and the subgraph selects some of the left and some of the right nodes,
and some of the edges. There will be no edges between two left nodes or between two right nodes.
1
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Common mistakes: Saying that a graph must be connected to be bipartite., or saying that if the
subgraph selects only nodes on one side, then it is not bipartite. If a graph has no edges, then it is
trivially bipartite. Another common mistake: Saying that if a graph has no circuits then it cannot
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 Spring '11
 Estie

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