  Given an $n$-vertex graph $G$ and two positive integers $d,k \in \mathbb{N}$,
the ($d,kn$)-differential coloring problem asks for a coloring of the vertices
of $G$ (if one exists) with distinct numbers from 1 to $kn$ (treated as
\emph{colors}), such that the minimum difference between the two colors of any
adjacent vertices is at least $d$. While it was known that the problem of
determining whether a general graph is ($2,n$)-differential colorable is
NP-complete, our main contribution is a complete characterization of bipartite,
planar and outerplanar graphs that admit ($2,kn$)-differential colorings. For
practical reasons, we consider also color ranges larger than $n$, i.e., $k >
1$. We show that it is NP-complete to determine whether a graph admits a
($3,2n$)-differential coloring. The same negative result holds for the
($\lfloor 2n/3 \rfloor, 2n$-differential coloring problem, even in the case
where the input graph is planar.
