The problem is: Find all natural numbers $n$ for which edges of a complete graph $K_n$ can be colored red and blue so that each vertex of a complete graph has an equal number of red and blue edges?
So I have already solved 4-5 problems with complete graphs and I am familiar with them. Complete graph with $n$ vertices has $m = n(n-1)/2$ edges and the degree of each vertex is $n-1$.
Because each vertex has an equal number of red and blue edges that means that $n-1$ is an even number $ \implies n$ has to be an odd number. Now possible solutions are $1, 3, 5, 7, 9, 11 ..$
What i did next is basically i drew complete graphs for some of these cases to see what happens. I found out that $5, 9$ are solutions but I don't know whether there is a formula or a way to determine all these numbers? Obviously i can't draw them all.
Example for $K_5$