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Chapter 13
15. We must find the chromatic number of the graph
shown. Note that 2,3,6,11 determine a subgraph
isomorphic to
1C
4 ,
so we need at least four colors. Let
us select
R, W, B,
G for these vertices, respectively.
We try to color the remaining vertices with four
colors. Note that 9 is joined to 2, 6 and 11, so it must
be
W.
Since 4 and 7 are not joined to 3 or 9, we may
color this
W
as well. Since 1 and 12 are not joined to
2 or to each other, we may color them
R.
Since 5 and
8 are not joined to 6 nor to each other, we color them
B.
Finally, 10 can be colored G.
Wg
379
R
12
Since
X(g)
=
4, the prisoners can be divided as follows: {I, 2, 12}, {3, 4,7, 9}, {5, 6, 8}, {1O, 11}.
16. The lineofsight graph
9
will have chromatic number 3. To
see why, note that
N
3 ,
N7
and
Ns
must be assigned different
colors, say
R, B,
G, respectively. Then
N2
and
N5
can be
colored G, and
N1
and
N4
can be colored
B.
Finally,
N6
can
be colored
R.
The partition of nets is
{Nl. N4, N7},
{N
2
,N
5
,Ns}.
{N
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This note was uploaded on 11/08/2010 for the course MATH discrete m taught by Professor Any during the Summer '10 term at FSU.
 Summer '10
 any
 Graph Theory

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