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
,N
s}.
{N
3
,N
6}.
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 Graph Theory, Glossary of graph theory, Graph coloring, National secondary road, Regional road, ince

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