Longest paths in random Apollonian networks and largest $r$-ary subtrees of random $d$-ary recursive trees

Let $r$ and $d$ be positive integers with $r<d$. Consider a random $d$-ary tree constructed as follows. Start with a single vertex, and in each time-step choose a uniformly random leaf and give it $d$ newly created offspring. Let ${\mathcal T}_t$ be the tree produced after $t$ steps. We show that there exists a fixed $δ<1$ depending on $d$ and $r$ such that almost surely for all large $t$, every $r$-ary subtree of ${\mathcal T}_t$ has less than $t^δ$ vertices. The proof involves analysis that also yields a related result. 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. In this way, one face is destroyed and three new faces are created. After $t$ steps, we obtain a random triangulated plane graph with $t+3$ vertices, which is called a random Apollonian network. We prove that there exists a fixed $δ<1$, such that eventually every path in this graph has length less than $t^δ$, which verifies a conjecture of Cooper and Frieze.

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