Combinatorics Through Guided Discovery, 2004a

188 D. Hints to Selected Problems
239. What does Problem 238 have to do with this question?
242.a. For each edge in F to connect two vertices of the same color, we must have
all the vertices in a connected component of the graph with vertex set V and
edge set F colored the same color.
242.c. How does the number you are trying to compute relate to the union of the
sets A i ?
243. One way to get a proper coloring of G − e is to start with a proper coloring of
G and remove e. But there are other colorings of G that become proper when
you remove e.
246. One approach would be to try to guess the result by doing a bunch of examples
and use induction to prove you are right. If you try this, what will you be
able to use to make the induction step work? There are other approaches as
well.
253.a. What do you want ϕ n ◦ ϕ −1 to be?
254. Ifσ i = σ j and i j, what can you conclude about ι?
256.b. What does it mean for one function to be the inverse of another one?
261. Once you know where the corners of the square go under the action of an
isometry, how much do you know about the isometry?
264. In how many ways can you choose a place to which you can move vertex
1? Having done that, in how many ways can you place the three vertices
adjacent to vertex 1?
265.a. In how many ways can you choose a place to which you can move vertex
1? Having done that, in how many ways can you place the three vertices
adjacent to vertex 1?
265.b. Why is it sufficient to focus on permutations that take vertex 1 to itself?
270. If a subgroup contains, say, ρ 3 and some flip, how many elements of D 4 must
it contain?
272. If the list (i σ(i) σ 2 (i) ... σ n (i)) does not have repeated elements but the
list (i σ(i) σ 2 (i) ... σ n (i) σ n+1 (i)) does have repeated elements, then which
element or elements are repeats?
277. The element k is either in a cycle by itself or it isn’t.