On Hamiltonian cycles in balanced $k$-partite graphs

For all integers $k$ with $k\ge 2$, if $G$ is a balanced $k$-partite graph on $n\ge 3$ vertices with minimum degree at least $$\lceil \frac{n}{2}\rceil +\lfloor \frac{n+2}{2\lceil \frac{k+1}{2}\rceil }\rfloor -\frac{n}{k}=\{\begin{array}{ll}\lceil \frac{n}{2}\rceil +\lfloor \frac{n+2}{k+1}\rfloor -\frac{n}{k}& :k\text{ odd }\\ \frac{n}{2}+\lfloor \frac{n+2}{k+2}\rfloor -\frac{n}{k}& :k\text{ even }\end{array},$$ 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 $\lceil \frac{n}{2}\rceil +\lfloor \frac{n+2}{2\lceil \frac{k+1}{2}\rceil }\rfloor -\frac{n}{k}+1$ suffices. This result is tight for all $k\ge 2$ and $n\ge 3$ divisible by $k$.