# Prove that a complete graph can be acyclic

Let $$G$$ be a complete graph. Prove that there always exists a way to assign $$n(n-1)/2$$ directed edges in a way that the graph will be acyclic (it will contain no directed circle). In other words, prove that every complete graph can be acyclic.

To clarify what I mean: Here's an example of one valid assignment for a 4-vertices complete graph:

It is acyclic, it is complete.

The only observation I have made is that the in-degrees for A-B-D-C respectively are 0,1,2,3; while the out-degrees for A-B-D-C respectively are 3,2,1,0, but don't really know whether it helps me to prove this.

Any clues?

Thank you in advance and sorry if some of the terms I have used aren't accurate, I'm not taking this course in English.

## 1 Answer

Assign the numbers $1,\ldots,n$ to the vertices, and for $i\ne j$, let an arrow point from vertex $i$ to vertex $j$ if and only if $i<j$. That's acyclic.