Digraph Automorphisms: Counting Edge Colorings

[Monte_Carlo]

Hello,

I have the following abstract algebra problem. It has to do with digraph automorphisms.

You're given a digraph G with vertices V(G) = {x, y, z, w} and edges E(G) = { (x, y), (x, z), (x, w) }. How many essentially different ways are there to color the edges of G using the following colors:

1) only white
2) both white and blue, each at least once
3) green, red and blue, each at least once
4) all red, all blue or some red and some blue

For 1), I think the answer is 1
For 2), I think the answer is 2 but it disagrees with the answer in the book.

Please give me some direction and opinion, I don't think just answers will be useful.

Thanks,

Monte

 

[Monte_Carlo]

Since it's the first time I've posted a request for homework assistance, I realize I might have made several mistakes which would explain why nobody has proffered an attempt at solution. For one, I'm not sure if this section of the forum is the right one. I've seen descriptions of other sections, for example one for abstract algebra. However, there is an instruction not to post homework problems there. The problem I'm asking about is not, strictly speaking, a homework problem, but it is one at the end of a textbook chapter.

I'd be happy to hear if there is any other additional information that would be expected from me to elicit an attempt at solution from anybody at this forum.

Thanks,

Monte