Distance Coloring.

Given a graph G = (V,E), a (d,k)-coloring is a function from the vertices V to colors {1,2,...,k} such that any two vertices within distance d of each other are assigned dierent colors. We determine the complexity of the (d,k)-coloring problem for all d and k, and enumerate some interesting properties of (d,k)-colorable graphs. Our main result is the discovery of a dichotomy between polynomial and NP-hard instances: for fixed d 2, the distance coloring problem is polynomial time for k b 3d 2 c and NP-hard for k > b 3d 2 c.