Let $D = (V, A)$ be a directed graph with minimum outdegree $\delta$ and maximum indegree $\Delta$. Prove (using Lovasz local lemma) that, if $k \le 1/(1+\ln(1+\delta\Delta))$, then $D$ contains a directed cycle of length divisible by $k$.
This question was asked in one of my class tests. My approach was that I was taking a subset $V'$ of vertices of length $p\times k$, $p$ is some constant greater than $0$ and creating a permutation sequence of these vertices. Then I was finding the probability of this sequence being a cycle. I am sure if I'm using the indegree and outdegree correctly also how to use Lovasz local lemma in this?