#### Date of Original Version

5-16-2014

#### Type

Article

#### Rights Management

This is the accepted version of the article which has been published in final form at http://dx.doi.org/10.1002/rsa.20576

#### Abstract or Description

We consider the height of random k-trees and k-Apollonian networks. These random graphs are not really trees, but instead have a tree-like structure. The height will be the maximum distance of a vertex from the root. We show that w.h.p. the height of random k-trees and k-Apollonian networks is asymptotic to c log t, where t is the number of vertices, and c = c(k) is given as the solution to a transcendental equation. The equations are slightly different for the two types of process. In the limit as k → ∞ the height of both processes is asymptotic to log t/(k log 2).

#### DOI

10.1002/rsa.20576

#### Published In

Random Structures and Algorithms, 45, 4, 675-702.