Combinatorics Through Guided Discovery, 2004a

124 6. Groups acting on sets
Problem 305. In how many ways may we paint the faces of a cube with two
colors of paint? What if both colors must be used? (h)
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Problem 306. In how many ways may we color the edges of a (regular)
(2n +1)-gon free to move around in the plane (so it cannot be flipped) if we
use red n times and blue n +1times? If this is a number you have seen
before, identify it. (h)
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Problem 307. In how many ways may we color the edges of a (regular)
(2n +1)-gon free to move in three-dimensional space so that n edges are
colored red and n +1edges are colored blue. Your answer may depend on
whether n is even or odd.
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Problem 308. (Not unusually hard for someone who has worked on chromatic
polynomials.) How many different proper colorings with four colors
are there of the vertices of a graph which is cycle on five vertices? (If we get
one coloring by rotating or flipping another one, they aren’t really different.)
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Problem 309. How many different proper colorings with four colors are
there of the graph in Figure 6.2.4? Two graphs are the same if we can
redraw one of the graphs, not changing the vertex set or edge set, so that it
is identical to the other one. This is equivalent to permuting the vertices in
some way so that when we apply the permutation to the endpoints of the
edges to get a new edge set, the new edge set is equal to the old one. Such
a permutation is called an automorphism of the graph. Thus two colorings
are different if there is no automorphism of the graph that carries one to the
other one.
1 2
6
3
5
4
Figure 6.2.4: A graph on six vertices.
(h)