I am being asked this:
Prove that it is impossible to have a group of 55 people at a party such that each one is a friend with exactly five of the others in the group.
I don't know if my reasoning is okay. I just want to check it here.
So, I like to think like this...
We have 55 people, so that means we have 55 vertices. Each of them have exactly 5 friends, which means they have 5 edges.
The theory says:
The number of vertices with an odd degree must be even
So, 55 is not even while it has an odd degree (5), thus: this is impossible.
Am I correct? Am I also correct if I'd say that this would be possible if there were 56 people with 5 edges?