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?