The chromatic and clique numbers of random scaled sector graphs

Random scaled sector graphs were introduced as a generalization of random geometric graphs to model networks of sensors using optical communication. In the random scaled sector graph model vertices are placed uniformly at random into the [0, 1]2 unit square. Each vertex i is assigned uniformly at random sector Si, of central angle αi, in a circle of radius ri (with vertex i as the origin). An arc is present from vertex i to any vertex j, if j falls in Si. In this work, we study the value of the chromatic number χ(Gn), directed clique number ω(Gn), and undirected clique number ω2 (Gn) for random scaled sector graphs with n vertices, where each vertex spans a sector of α degrees with radius rn = √ln n/n. We prove that for values α π w.h.p., χ(Gn) and ω2 (Gn) are Θ (ln n), being the same for random scaled sector and random geometric graphs, while ω(Gn) is Θ(ln n/ln ln n).