The Maximum k-Differential Coloring Problem

Given an n-vertex graph G and two positive integers d, k ∈ 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 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, n)-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 (b2n/3c, 2n)-differential coloring problem, even in the case where the input graph is planar.