Digraph with no even cycle

Question

Let D be a digraph with no even directed cycle. Prove that D has at most one kernel.

Could someone please help with a proof?

Answer 1

Suppose you have two different kernels $A$ and $B$. Here are a couple of hints:

1. $A \setminus B$ and $B \setminus A$ must be non empty.
2. The induced graph on $(A \setminus B) \cup (B \setminus A)$ is bipartite, and all its vertices have indegree at least 1.

This is enough to conclude that there is an even directed cycle, i.e. a contradiction.