Integer-Programming Bounds on Pebbling Numbers of Cartesian-Product Graphs

  • Franklin Kenter
  • Daphne SkipperEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11346)


Graph pebbling, as introduced by Chung, is a two-player game on a graph G. Player one distributes “pebbles” to vertices and designates a root vertex. Player two attempts to move a pebble to the root vertex via a sequence of pebbling moves, in which two pebbles are removed from one vertex in order to place a single pebble on an adjacent vertex. The pebbling number of a simple graph G is the smallest number \(\pi _G\) such that if player one distributes \(\pi _G\) pebbles in any configuration, player two can always win. Computing \(\pi _G\) is provably difficult, and recent methods for bounding \(\pi _G\) have proved computationally intractable, even for moderately sized graphs.

Graham conjectured that the pebbling number of the Cartesian-product of two graphs G and H, denoted \(G\,\square \,H\), is no greater than \(\pi _G \pi _H\). Graham’s conjecture has been verified for specific families of graphs; however, in general, the problem remains open.

This study combines the focus of developing a computationally tractable method for generating good bounds on \(\pi _{G \,\square \, H}\), with the goal of providing evidence for (or disproving) Graham’s conjecture. In particular, we present a novel integer-programming (IP) approach to bounding \(\pi _{G \,\square \, H}\) that results in significantly smaller problem instances compared with existing IP approaches to graph pebbling. Our approach leads to a sizable improvement on the best known bound for \(\pi _{L \,\square \, L}\), where L is the Lemke graph. \(L\,\square \, L\) is among the smallest known potential counterexamples to Graham’s conjecture.


  1. 1.
    Asplund, J., Hurlbert, G., Kenter, F.: Pebbling on graph products and other binary graph constructions. Australas. J. Comb. 71(2), 246–260 (2017)MathSciNetGoogle Scholar
  2. 2.
    Chung, F.: Pebbling in hypercubes. SIAM J. Discrete Math. 2(4), 467–472 (1989)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Cook, W., Koch, T., Steffy, D.E., Wolter, K.: An exact rational mixed-integer programming solver. In: Günlük, O., Woeginger, G.J. (eds.) IPCO 2011. LNCS, vol. 6655, pp. 104–116. Springer, Heidelberg (2011). Scholar
  4. 4.
    Cranston, D.W., Postle, L., Xue, C., Yerger, C.: Modified linear programming and class 0 bounds for graph pebbling. J. Comb. Optim. 34(1), 114–132 (2017). Scholar
  5. 5.
    Cusack, C.A., Green, A., Bekmetjev, A., Powers, M.: Graph pebbling algorithms and Lemke graphs. SubmittedGoogle Scholar
  6. 6.
    Elledge, S., Hurlbert, G.H.: An application of graph pebbling to zero-sum sequences in Abelian groups. Integers 5, A17 (2005)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Feng, R., Kim, J.Y.: Pebbling numbers of some graphs. Sci. China Ser. A Math. 45(4), 470–478 (2002). Scholar
  8. 8.
    Gurobi Optimization Inc.:
  9. 9.
    Herscovici, D.S.: Graham’s pebbling conjecture on products of cycles. J. Graph Theory 42(2), 141–154 (2003). Scholar
  10. 10.
    Herscovici, D.S., Higgins, A.W.: The pebbling number of \({C_5} \times {C_5}\). Discrete Math. 187(1), 123–135 (1998). Scholar
  11. 11.
    Hurlbert, G.: A linear optimization technique for graph pebbling. arXiv e-prints, January 2011Google Scholar
  12. 12.
    Hurlbert, G.: Graph pebbling. In: Gross, J.L., Yellen, J., Zhang, P. (eds.) Handbook of Graph Theory, pp. 1428–1449. Chapman and Hall/CRC, Kalamazoo (2013)Google Scholar
  13. 13.
    Lemke, P., Kleitman, D.: An addition theorem on the integers modulo n. J. Number Theory 31(3), 335–345 (1989). Scholar
  14. 14.
    Milans, K., Clark, B.: The complexity of graph pebbling. SIAM J. Discrete Math. 20(3), 769–798 (2006). Scholar
  15. 15.
    Snevily, H.S., Foster, J.A.: The 2-pebbling property and a conjecture of Graham’s. Graphs Comb. 16(2), 231–244 (2000). Scholar
  16. 16.
    Wang, S.: Pebbling and Graham’s conjecture. Discrete Math. 226(1), 431–438 (2001). Scholar
  17. 17.
    Wang, Z., Zou, Y., Liu, H., Wang, Z.: Graham’s pebbling conjecture on product of thorn graphs of complete graphs. Discrete Math. 309(10), 3431–3435 (2009). Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.U.S. Naval AcademyAnnapolisUSA

Personalised recommendations