Question About Ore's thm and Dirac's thm on digraph

Question

I should prove that in digraph, Ore's thm implies Dirac's thm.

It means D: simple digraph with n vertices, and

(outdeg $v$ + indeg $w$ $>=n$) s.t $v$ and $w$ are vertices in D , and $v$ is not adjacent to $w$, then D is Hamiltonian

implies

outdeg $v$ $>=\frac {n}{2}$ and indeg $v$ $>=$ $\frac{n}{2}$ for each vertex of D then D is Hamiltonian.

I tried outdeg $v$ + indeg $w$ $>= n$ and outdeg $w$ +indeg $v$ $>=n$

But I failed... what can I do??

Answer 1

I think you got the logic backwards (or maybe I misinterpreted the question). Suppose we want to show that Ore's theorem implies Dirac's theorem. This means we want to prove Dirac's theorem assuming Ore's theorem. In other words:

• We are given a directed graph $D = (V,A)$ with the property that $\text{outdeg}(v) \geq \frac{n}{2}$ and $\text{indeg}(v) \geq \frac{n}{2}$ holds for every vertex $v \in V$.
• Now we must somehow prove that $D$ contains a Hamiltonian cycle. We don't have to start from scratch though: we are allowed to use Ore's theorem. Can we show that $D$ meets the hypothesis of Ore's theorem?