On longest paths and diameter in random Apollonian networks

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). See http://www.math.cmu.edu/~ctsourak/ran.html for an example.

We prove that the diameter of a RAN is asymptotic to \(c \log(n)\) in probability, where \(c \approx 1.668\) is the solution of an explicit equation. The proof adapts a technique of Broutin and Devroye for estimating the height of random trees.

We also prove that there exists a fixed \(s<1\), such that eventually every self-avoiding walk in this graph has length less than \(n^s\), which verifies a conjecture of Cooper and Frieze. Using a similar technique, we show that if \(r < d\) are fixed constants, then every r-ary subtree of a random d-ary recursive tree on n vertices has less than \(n^b\) vertices, for some \(b=b(d,r)<1\).

Based on joint work with A. Collevecchio, E. Ebrahimzadeh, L. Farczadi, P. Gao, C. Sato, N. Wormald, and J. Zung.