Surprising identities for the greedy independent set on Cayley trees

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 $G_n$ . 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 that is of independent interest.

Greedy maximal independent sets via local limits

“…To the best of our knowledge, this intriguing fact was not previously known. Recently, after a conference version of this paper was published, Contat [17] proved a much stronger statement concerning the cardinality of the random greedy independent set in uniform random trees, showing that it has essentially the same law as its complement. In a newer version of her paper, she obtained the exact distribution of the size of the random greedy independent set, showing that it has the same distribution as the number of vertices at even height in a uniformly sampled rooted random tree.…”