On Longest Paths and Diameter in Random Apollonian Networks

We consider the following iterative construction of a random planar triangulation. Start with a triangle embedded in the plane. In each step, choose a bounded face uniformly at random,add a vertex inside that face and join it to the vertices of the face. After n-3 steps, we obtain a randomtriangulated plane graph with n vertices, which is called a Random Apollonian Network (RAN). Weshow that asymptotically almost surely (a.a.s.) a longest path in a RAN has length o(n), refuting aconjecture of Frieze and Tsourakakis. We also show that a RAN always has a cycle (and thus a path)of length (2n -5)(log 2/ log 3), and that the expected length of its longest cycles (and paths) is ohm(n(0.88)). Finally, we prove that a.a.s. the diameter of a RAN is asymptotic to c log n, where c = 1.668 is the solution of an explicit equation. (C) 2014 Wiley Periodicals, Inc. Random Struct. Alg., 45, 703-725, 2014