For which values of $m$ is $m$-partition hard? Consider the following reduction: $m$-colouring to $m$-partition.

Define an $m$-partition as: Given an undirected graph $G = (V, E)$ and an integer $j$. Does there exist a partition of the vertices into $m$ parts $\{V_1, V_2, \ldots , V_m\}$ such that at least $j$ of the edges have their endpoints in different parts of the partition?

I was able to come up with the reduction, but now I'm stuck at this question:

What I need to find is the values of $m$ where the reduction implies that $m$-Partition is NP-complete?
The $1$-partition problem is trivial, since there is only one possible partition.

The $2$-partition problem is the same as MAX-CUT, which is known to be NP-complete.

For $m &gt; 2$, the $m$-partition problem is NP-complete by reduction from $m$-coloring, which is NP-complete for that range of $m$.