Characterizing Winning Positions in the Impartial Two-Player Pebbling Game on Complete Graphs
96 Frelinghuysen Road
Piscataway, NJ 08854
University Scholars Program
Pennsylvania Leadership Charter School
1585 Paoli Pike
West Chester, PA 19380
Department of Mathematics & Statistics
800 Lancaster Avenue
Villanova, PA 19085
Tony W. H. Wong
Department of Mathematics
Kutztown University of Pennsylvania
15200 Kutztown Road
Kutztown, PA 19530
A pebbling game on a simple graph consists of moves that remove two
pebbles from a vertex of the graph and add one pebble to an adjacent
vertex. We consider an impartial two-player pebbling game, where the
winner of the game is the last player to make an allowable pebbling
move. In this paper, we characterize the winning positions when this game
is played on a complete graph Kn
with at least n + 2 pebbles if n ≥ 5 is
odd and at least n + 7 pebbles if n ≥ 6 is even.
is independent of how the pebbles are initially distributed on the
vertices and depends only on the parity of the total number of pebbles
used in the game.
Full version: pdf,
(Concerned with sequence
Received March 25 2021; revised version received May 4 2021.
Published in Journal of Integer Sequences,
May 18 2021.
Journal of Integer Sequences home page