  For all integers $k$ with $k\geq 2$, if $G$ is a balanced $k$-partite graph
on $n\geq 3$ vertices with minimum degree at least \[
\left\lceil\frac{n}{2}\right\rceil+\left\lfloor\frac{n+2}{2\lceil\frac{k+1}{2}\rceil}\right\rfloor-\frac{n}{k}=\begin{cases}
\lceil\frac{n}{2}\rceil+\lfloor\frac{n+2}{k+1}\rfloor-\frac{n}{k} & : k \text{
odd }\\ \frac{n}{2}+\lfloor\frac{n+2}{k+2}\rfloor-\frac{n}{k} & : k \text{ even
} \end{cases}, \] then $G$ has a Hamiltonian cycle unless $k=2$ and 4 divides
$n$, or $k=\frac{n}{2}$ and 4 divides $n$. In the case where $k=2$ and 4
divides $n$, or $k=\frac{n}{2}$ and 4 divides $n$, we can characterize the
graphs which do not have a Hamiltonian cycle and see that
$\left\lceil\frac{n}{2}\right\rceil+\left\lfloor\frac{n+2}{2\lceil\frac{k+1}{2}\rceil}\right\rfloor-\frac{n}{k}+1$
suffices. This result is tight for all $k\geq 2$ and $n\geq 3$ divisible by
$k$.
