Painting the edges of an $n$-gonal prism with 3 colors all the edges of each vertex have different colors if $n= 2018$ and $n= 2019$?

Question

Is it possible to paint the edges of a n-gonal prism with 3 colors so that each face has all 3 colors and all the edges of each vertex have different colors if a)n= 2018; b)n= 2019?
We can look at it as n=odd and n= even cases.
If n=odd:
Let us say one of the parallel faces has colour C1, then the three adjacent quadrilateral can be either C2 or C3. And this is not possible without repeating the colour. Hence, n=2019 is not possible.
If n=even:
Then let one of the parallel faces be C1, the 4 quadrilaterals can be coloured with C2 and C3 and the opposite face with C1. Hence, n=2018 is possible.
Now, this explanation doesn't take the colour of the edges in account. If we take them into account, how would we proceed then?

Siddhartha · Answer

Yes. I remember now, after a point, I had to stop looking at odd/even and this multiple condition worked. You’re right.

Ross Millikan · Answer

When the bases have an even number of sides, paint all the edges connecting the two bases with C1.  You can then paint alternating sides C2 and C3 to get a coloring that satisfies your requirement.  You cannot do this when the bases have an odd number of sides.
When the bases have a number of sides that is a multiple of $3$, paint one base with a cycle of C1, C2, C3.  Each edge then has the third color for its vertex and you can paint the other base with the cycle again.

Painting the edges of an $n$-gonal prism with 3 colors all the edges of each vertex have different colors if $n= 2018$ and $n= 2019$?

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