On Maximum Differential Coloring of Planar Graphs

We study the maximum differential coloring problem, where the vertices of an n-vertex graph must be labeled with the numbers 1, ..., n such that the minimum difference between the two labels of any adjacent vertices is maximized. As it is NP-hard to find the optimal labels for general graphs, we consider special sub-classes: caterpillars, spiders, and extended stars. We first prove new upper bounds for maximum differential coloring for spiders and regular caterpillars. Using these new bounds, we prove that the Miller-Pritikin [18] labeling scheme for forests is optimal for regular caterpillars and for spider graphs. On the other hand, we give examples of general caterpillars where the Miller-Pritikin algorithm behaves very poorly. We present an alternative algorithm for general caterpillars which achieves reasonable results even in worst case. Finally we present new optimal labeling schemes for regular caterpillars and for sub-classes of spider graphs.