Séminaire Lotharingien de Combinatoire, 86B.85 (2022), 12 pp.
Colin Defant, Mike Joseph, Matthew Macauley and Alex McDonough
Torsors From Toggling Independent Sets
Abstract.
In this paper, we consider the problem of toggling independent sets in cycle graphs. In each orbit, we find an infinite abelian "snake group" that acts simply transitively on the "live entries". This allows us to characterize a number of combinatorial properties of the dynamics by studying the topological covering maps between this torsor and finite quotients. We also characterize the orbits via solutions to a three-variable Diophantine equation. Preliminary work has found other toggle actions where the live entries are a torsor for a group, suggesting that this work is a special case of a more general framework, and posing the question of when this phenomenon arises and why.
Received: November 25, 2021.
Accepted: March 4, 2022.
Final version: April 1, 2022.
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