Vol. 12, No. 2, 2019

Download this article
Download this article For screen
For printing
Recent Issues

Volume 13
Issue 3, 361–539
Issue 2, 181–360
Issue 1, 1–180

Volume 12, 8 issues

Volume 11, 5 issues

Volume 10, 5 issues

Volume 9, 5 issues

Volume 8, 5 issues

Volume 7, 6 issues

Volume 6, 4 issues

Volume 5, 4 issues

Volume 4, 4 issues

Volume 3, 4 issues

Volume 2, 5 issues

Volume 1, 2 issues

The Journal
About the Journal
Editorial Board
Editors’ Interests
Submission Guidelines
Submission Form
Policies for Authors
Ethics Statement
ISSN: 1944-4184 (e-only)
ISSN: 1944-4176 (print)
Author Index
Coming Soon
Other MSP Journals
This article is available for purchase or by subscription. See below.
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.

PDF Access Denied

However, your active subscription may be available on Project Euclid at

We have not been able to recognize your IP address as that of a subscriber to this journal.
Online access to the content of recent issues is by subscription, or purchase of single articles.

Please contact your institution's librarian suggesting a subscription, for example by using our journal-recom­mendation form. Or, visit our subscription page for instructions on purchasing a subscription.

You may also contact us at contact@msp.org
or by using our contact form.

Or, you may purchase this single article for USD 30.00:

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