If a graph $G$ has no loops or parallel edges, and if the number of vertices $n$ in the graph is $n\geq 3$, then graph $G$ is Hamiltonian if:

$(i)$ $\deg(v)\geq \dfrac{n}{3}$ for each vertex $v$

$(ii)$ $\deg(v)+\deg(w)\geq n$ whenever $v$ and $w$ are not connected by an edge.

$(iii)$ $E(G)\geq \dfrac{1}{3}(n-1)(n-2)+2$

Choose the correct answer from the code given below:

Statement $(i)$ is not a standard sufficient condition for Hamiltonian graph.

By Dirac's theorem, the sufficient condition is

$\deg(v)\geq \dfrac{n}{2}$

for every vertex $v$.

So, statement $(i)$ is false.

Statement $(ii)$ is true by Ore's theorem.

Ore's theorem says that if for every pair of non-adjacent vertices $v$ and $w$,

$\deg(v)+\deg(w)\geq n$

then the graph is Hamiltonian.

Statement $(iii)$ is not the standard sufficient condition here.

Therefore, only statement $(ii)$ is correct.