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 random triangulated plane graph with n vertices, which is called a Random Apollonian Network (RAN). We show that asymptotically almost surely (a.a.s.) a longest path in a RAN has length o ( n ), refuting a conjecture of Frieze and Tsourakakis. We also show that a RAN always has a cycle (and thus a path) of length( 2 n − 5 )   log   ⁡ 2 /   log   ⁡ 3, and that the expected length of its longest cycles (and paths) is Ω ( 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. © 2014 Wiley Periodicals, Inc. Random Struct. Alg., 45, 703–725, 2014