Let $G$ be a graph with $n+k$ vertices such that $n$ vertices have degree at least $\frac{n+k}{2}$ and the remaining $k$ vertices have degree at least $k+1$. Show that G is Hamiltonian.
Using Diracs Theorem I need to show that the minimum degree of G is bigger that $\frac{n+k}{2}$. which implies I have to show that $k+2 \geq n$.
It is clear that each of the $k$ vertices have at least 2 neighbours of among the $n$ vertices.
How can I use this to prove that G is Hamiltonian?