K - The Perfect Tour

There is a grid of $N \cdot M$ cells. It is guaranteed that $N \cdot M$ is even. Exactly half of the cells, i.e. $\frac{N \cdot M}{2}$ are coloured white. The rest are coloured black. Construct $2$ directed tournament$^\dagger$ graphs: one on the black cells, and another on the white cells. There are a total of $C(\frac{N \cdot M}{2}, 2)^{\ddagger}$ edges in each tournament graph. Thus, a total o