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Surprising identities for the greedy independent set on Cayley trees

Part of: Graph theory Markov processes Algorithms - Computer Science

Published online by Cambridge University Press:  25 August 2022

Alice Contat
Alice Contat*
Affiliation:
Université Paris-Saclay
*
*Postal address: Institut Mathématiques d’Orsay, Bâtiment 307, Université Paris-Saclay, 91405 Orsay, France. Email address: alice.contat@universite-paris-saclay.fr
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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.

Keywords

Cayley treesindependent setgreedy

MSC classification

Primary: 05C80: Random graphs
Secondary: 68W20: Randomized algorithms 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)
Type
Original Article
Information
Journal of Applied Probability , Volume 59 , Issue 4 , December 2022 , pp. 1042 - 1058
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Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Applied Probability Trust

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