Peg solitaire on line graphs

Document Type : Research Paper


1 Institute of Mathematics and Applied Computer Science, University of Hildesheim, Samelsonplatz 1, 31141 Hildesheim, Germany

2 Institute of Mathematics, Freie Universita╠łt Berlin, Arnimallee 3, 14195 Berlin, Germany


In $2011$, Beeler and Hoilman generalized the game of peg solitaire to arbitrary connected graphs. Since then peg solitaire and related games have been considered on many graph classes. One of the main goals is the characterization of solvable graphs. To this end, different graph operations, such as joins and Cartesian products, have been considered in the past. In this article, we continue this venue of research by investigating line graphs. Instead of playing peg solitaire on the line graph $L(G)$ of a graph $G$, we introduce a related game called stick solitaire and play it on $G$. This game is examined on several well-known graph classes, for example complete graphs and windmills. In particular, we prove that most of them are stick-solvable. We also present a family of graphs which contains unsolvable graphs in stick solitaire. Naturally, the Fool's stick solitaire number is an object of interest, which we compute for the previously considered graph classes.


Main Subjects

[1] G. I. Bell, Solving triangular peg solitaire, J. Integer Seq., 11 (2008) 22 pp.
[2] R. A. Beeler, A. D. Gray and D. P. Hoilman, Constructing solvable graphs in peg solitaire, Bull. Inst. Combin. Appl., 66 (2012) 89–96.
[3] R. A. Beeler, H. Green and R. T. Harper, Peg solitaire on caterpillars, Integers, 17 (2017) 14 pp.
[4] R. A. Beeler and D. P. Hoilman, Peg solitaire on graphs, Discrete Math., 311 (2011) 2198–2202.
[5] R. A. Beeler and D. P. Hoilman, Peg solitaire on the windmill and the double star graphs, Australas. J. Combin., 52 (2012) 127–134.
[6] R. A. Beeler and T. K. Rodriguez, Fool’s solitaire on graphs, Involve, 5 (2012) 473–480.
[7] R. A. Beeler and C. A. Walvoort, Peg solitaire on trees with diameter four, Australas. J. Combin., 63 (2015) 321–332.
[8] J.-H. de Wiljes and M. Kreh, Peg solitaire on banana trees, Bull. Inst. Combin. Appl., 90 (2020) 63–86.
[9] J.-H. de Wiljes and M. Kreh, Path-stick solitaire on graphs, Theory Appl. Graphs, 9 no. 2 (2022) 11 pp.
[10] J. Engbers and C. Stocker, Reversible peg solitaire on graphs, Discrete Math., 338 (2015) 2014–2019.
[11] M. Kreh and J.-H. de Wiljes, Peg solitaire on Cartesian products of graphs, Graphs Comb., 37 (2021) 907–917.
[12] S. Loeb and J. Wise, Fool’s solitaire on joins and Cartesian products of graphs, Discrete Math., 338 (2015) 66–71.
[13] N. Matsumoto, Peg solitaire on graphs with large maximum degree, Asian-Eur. J. Math., 15 (2022) 9 pp.
Volume 13, Issue 3 - Serial Number 3
September 2024
Pages 257-277
  • Receive Date: 16 November 2021
  • Revise Date: 27 March 2023
  • Accept Date: 02 August 2023
  • Published Online: 01 September 2024