Slither code and the independence number of a random tree

TL;DR

This paper provides a simple probabilistic characterization of the independence number, matching number, and path cover number of a uniformly random labeled tree, using modifications of the Prüfer code.

Contribution

It introduces a novel probabilistic method to determine key graph parameters of random trees via a simple die-rolling process and bijective proofs.

Findings

Distribution of independence number matches the smallest number of die throws for a certain coverage.

Distribution of matching number is related to the number of distinct die outcomes after a certain number of throws.

Path cover number distribution is similarly characterized.

Abstract

We give a simple characterisation of the distribution of the independence number, and equivalently the matching number, of a random tree on $n$ labelled vertices chosen uniformly among the $n^{n-2}$ such trees: Roll an $n$-sided die repeatedly, and let $\alpha$ be the smallest number such that after $\alpha$ throws, at least $n-\alpha$ distinct numbers have occurred. Then $\alpha$ has the same distribution as the independence number, and $n-\alpha$ has the same distribution as the matching number. We obtain a similar characterisation of the path cover number. The proofs are bijective and based on modifications of the Pr\"ufer code.