On Coloring the Square of Unit Disk Graph (DRAFT)

A graph G is a Unit Disk (UD) graph if there is an assignment of unit disks centered at its vertices such that there is an edge in G iff the corresponding unit disks intersect. The square G of G is defined as a graph on the same vertex set as G and having edges between pairs of vertices with graph distance at most two in G. Our results on the chromatic number of G are motivated by an application for wireless sensor networks. Namely, if G, an UD, is an underlying graph of a wireless sensor network, the square of G, G, reflects possible conflicts in transmission between sensors in the network. An upper bound on the chromatic number of G implies directly an upper bound on the frame length in the Time Division Multi-Allocation (TDMA) used for transmission scheduling in the sensor network. The main result of the present paper shows that the chromatic number of G is linear in ω(G) and is at most 13ω(G), where ω(G) is the independence number of G. We also present an efficient algorithm to find a coloring corresponding to the upper bound.