Abstract We study the maximum differential coloring problem, where the vertices of an n-vertex graph must be labeled with distinct numbers ranging from 1 to n, so that the minimum absolute difference between two labels of any two adjacent vertices is maximized. As the problem is NP-hard for general graphs , we consider planar graphs and subclasses thereof. We prove that the maximum differential coloring problem remains NP-hard, even for planar graphs. We also present tight bounds for regular caterpillars and spider graphs. Using these new bounds, we prove that the Miller–Pritikin labeling scheme  for forests is optimal for regular caterpillars and for spider graphs.