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370
16. [BB] Three exam periods are
required since this is the chromatic
number of the graph.
17. We draw a graph whose vertices correspond to courses and
an edge means that the two corresponding exams should
not be scheduled at the same time (because there are stu
dents taking each course). The chromatic number of the
graph is 4 (a 4coloring is shown). This is best possible be
cause the graph contains
K4 (E, M, F, P).
The minimum
number of exam periods to avoid conflicts is four.
3
R
4
5
W
R
Solutions to Exercises
6
B
WE~MR
pY
F
G
y
B
C
R
18. Our proof does
not
work for four colors. We show why by analyzing our proof of The Five Color
Theorem. Let v be a vertex of degree five, adjacent to vertices Vb V2, V3, V4, V5, and let
go
=
g
"
v.
The induction hypothesis would say that
X(Qo)
::; 4. Hence two of the Vi would be assigned the same
color. Suppose the colors
R, W, B,
G were assigned as in the
figure and also that VIV5 and V2V4 were edges (as shown). Here
there is no path which alternates colors between VI and V3, but
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 Summer '10
 any
 Graph Theory

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