Vol. 12, No. 2, 2019

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Lights Out for graphs related to one another by constructions

Laura E. Ballard, Erica L. Budge and Darin R. Stephenson

Vol. 12 (2019), No. 2, 181–201
DOI: 10.2140/involve.2019.12.181

The Lights Out problem on graphs, in which each vertex of the graph is in one of two states (“on” or “off”), has been investigated from a variety of perspectives over the last several decades, including parity domination, cellular automata, and harmonic functions on graphs. We consider a variant of the Lights Out problem in which the possible states for each vertex are indexed by the integers modulo k. We examine the space of “null patterns” (i.e., harmonic functions) on graphs, and use this as a way to prove theorems about Lights Out on graphs that are related to one another by two main constructions.

graph theory, Lights Out
Mathematical Subject Classification 2010
Primary: 05C50, 05C69
Received: 15 December 2014
Revised: 6 September 2017
Accepted: 22 May 2018
Published: 8 October 2018

Communicated by Kenneth S. Berenhaut
Laura E. Ballard
Mathematics Department
Syracuse University
Syracuse, NY
United States
Erica L. Budge
Department of Mathematics
Hope College
Holland, MI
United States
Darin R. Stephenson
Department of Mathematics
Hope College
Holland, MI
United States