The cover time of random geometric graphs

We study the cover time of random geometric graphs. Let I(d) = [0, 1](d) denote the unit torus in d dimensions. Let D(x, r) denote the ball (disc) of radius r. Let Gamma(d) be the volume of the unit ball D(0,1) in d dimensions. A random geometric graph G = G(d, r, n) in d dimensions is defined as follows: Sample n points V independently and uniformly at random from I(d). For each point x draw a ball D(x,r) of radius r about x. The vertex set V(G) = V and the edge set E(G) = {{v,w} : w not equal v, w is an element of D(v, r)}. Let G(d, r, n), d >= 3 be a random geometric graph. Let c > 1 be constant, and let r = (c log n/(Gamma(d)n))(1/d). Then whp C-G similar to c log (c/c - 1) n log n.