Camel Up Cup: An AI board game tournament

I had the same problem in Mail / OS X 10.10.2.

After long searches for an answer, I solved the problem simply inserting again my passoword for the SMTP account.

Even though the stored password was correct, it didn't work until I inserted again the password. Go figure why it didn't work before !!

Updating to El Capitan, I had this same problem. When I checked my mail settings I noticed that my incoming port was changed to 143 from 993. I changed it back to 993 and turned on SSL and all worked again.

For 2., you want to use that the sum of the degrees is twice the number of edges, as Matt points out in the comments to the question. For 1., you should just construct a graph --- try putting the points in a circle, and try connecting each vertex to some of them that are near it on the circle. Hint: you'll have to do cases for if $c$ is even or odd.