Surprising identities for the greedy independent set on Cayley trees

TL;DR

This paper reveals a surprising symmetry in the distribution of the greedy independent set size on Cayley trees, linking it to the count of vertices at even height and providing exact laws and a Markovian construction.

Contribution

It introduces a novel symmetry in the distribution of greedy independent sets on Cayley trees and develops a new Markovian exploration method for these trees.

Findings

G_n has the same law as the number of vertices at even height in the tree

Exact law of G_n can be computed explicitly

A new Markovian exploration of Cayley trees is introduced

Abstract

We prove a surprising symmetry between the law of the size $G_n$ of the greedy independent set on a uniform Cayley tree $ \mathcal{T}_n$ of size $n$ and that of its complement. We show that $G_n$ has the same law as the number of vertices at even height in $ \mathcal{T}_n$ rooted at a uniform vertex. This enables us to compute the exact law of the $G_n$. We also give a Markovian construction of the greedy independent set, which highlights the symmetry of $G_n$ and whose proof uses a new Markovian exploration of rooted Cayley trees which is of independent interest.