Reduction from a m-coloring to a m-partition
By definition, a $m$-coloring is a function $f : V \to \{1,\dots, m\}$ such that $\forall u,v \in V \; f(u) = f(v) \implies (u,v) \not\in E$. This means that at least $|E|$ edges have their endpoints in different sets of the partition $\{V_1, \dots, V_m\}$ of $V$ with $V_i = \{ v \in V : f(v)=i \}$.

On the contrary, if there is a partition $\{V_1, \dots, V_m\}$ of $V$ such that at least $|E|$ edges have their endpoints in different sets, then the function $f : V \to \{1, \dots, k\}$, where $f(u)$ is the unique index $i_u \in \{1, \dots, m\}$ such that $u \in A_{i_u}$, satisfies $\forall u,v \in V \; f(u) = f(v) \implies (u,v) \not\in E$.

The above shows that the instance $G=(V,E)$ of $m$-coloring can be reduced (w.r.t. Karp reductions) to the instance $\langle G, |E| \rangle$ of $m$-partition.