Local 7-coloring for planar subgraphs of unit disk graphs

We study the problem of computing locally a coloring of an arbitrary planar subgraph of a unit disk graph. Each vertex knows its coordinates in the plane and can communicate directly with all its neighbors within unit distance. Using this setting, first a simple algorithm is given whereby each vertex can compute its color in a 9-coloring of the planar graph using only information on the subgraph located within at most 9 hops away from it in the original unit disk graph. A more complicated algorithm is then presented whereby each vertex can compute its color in a 7-coloring of the planar graph using only information on the subgraph located within a constant number (201, to be exact) of hops away from it.