Highest Trees of Random Mappings

Abstract

Using a symbolic method, we prove that the probability that the underlying graph of a random mapping of $n$ elements possesses a unique highest tree in the entire functional graph is $1 - \sqrt{\frac{\pi}{8}} n^{-1/2} + \mathcal{O}(n^{-1})$. The property of having a unique highest tree plays a crucial role in the solution of the famous Road Coloring Problem [Trahtman~2009]. We also generalize this result to $c$-branches (subtrees rooted at distance $c$ from the cycle core), namely, that for any constant $c>0$ the highest $c$-branch is unique, dominates the second highest by at least a height of 2, and supports a crown exceeding $\alpha$ times its root count (for any fixed constant $\alpha > 1$), with probability $1 - \Theta(n^{-1/2})$. The latter result is used in the author's paper establishing the $1-\mathcal{O}(1/n)$ bound for a random automaton being synchronizable.