Generalized Impartial Two-player Pebbling Games on K3 and C4

Generalized Impartial Two-player Pebbling Games on K₃ and C₄

Kayla Barker
Mathematics Program
Stockton University
101 Vera King Ferris Dr
Galloway, NJ 08205
USA

Mia DeStefano
Dept. of Mathematics and Statistics
Vassar College
124 Raymond Ave
Poughkeepsie, NY 12604
USA

Eugene Fiorini
DIMACS
Rutgers University
96 Frelinghuysen Road
Piscataway, NJ 08854
USA

Michael Gohn
Dept. of Mathematics and Computer Sci.
DeSales University
2755 Station Avenue
Center Valley, PA 18034
USA

Joe Miller
Department of Mathematics
Iowa State University
411 Morrill Rd
Ames, IA 50011
USA

Jacob Roeder
Dept. of Mathematics and Physics
Trine University
1 University Ave
Angola, IN 46703
USA

Tony W. H. Wong
Department of Mathematics
Kutztown University of Pennsylvania
15200 Kutztown Road
Kutztown, PA 19530
USA

Abstract:

In a variation on the pebbling game played on a simple graph, a (k + 1 : k)-pebbling move comprises removing k + 1 pebbles from a vertex and adding k pebbles to an adjacent vertex. We consider an impartial two-player game, where the winner of the game is the last player to make an allowable (k + 1 : k)-pebbling move. In this paper, we characterize the winning positions when the (k + 1 : k)-pebbling game is played on the complete graph K₃ and when the (2 : 1)-pebbling game is played on the cycle C₄.

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Received June 4 2023; revised versions received June 9 2023; May 14 2024. Published in Journal of Integer Sequences, June 3 2024.

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