Skip to main content Accessibility help

Signed Chip Firing Games and symmetric Sandpile Models on the cycles

  • Robert Cori (a1), Thi Ha Duong Phan (a2) and Thi Thu Huong Tran (a2)


We investigate the Sandpile Model and Chip Firing Game and an extension of these models on cycle graphs. The extended model consists of allowing a negative number of chips at each vertex. We give the characterization of reachable configurations and of fixed points of each model. At the end, we give explicit formula for the number of their fixed points.



Hide All
[1] Bak, P., Tang, C. and Wiesenfeld, K., Self-organized criticality : An explanation of 1/f noise. Phys. Rev. Lett. 59 (1987) 381284.
[2] Björner, A., Lovász, L. and Shor, P.W., Chip-firing games on graphs. Eur. J. Combin. 12 (1991) 283291.
[3] Cori, R. and Rossin, D., On the sandpile group of dual graphs. Eur. J. Combin. 21 (2000) 447459.
[4] Desel, J., Kindler, E., Vesper, T. and Walter, R., A simplified proof for the self-stabilizing protocol : A game of cards. Inf. Process. Lett. 54 (1995) 327328.
[5] Dhar, D., Ruelle, P., Sen, S. and Verma, D.-N., Algebraic aspects of abelian sandpile models. J. Phys. A 28 (1995) 805831.
[6] Formenti, E., Masson, B. and Pisokas, T., Advances in symmetric sandpiles. Fundam. Inf. 76 (2007) 91112.
[7] Goles, E. and Kiwi, M.A., Games on line graphes and sand piles. Theoret. Comput. Sci. 115 (1993) 321349.
[8] Goles, E., Morvan, M. and Phan, H.D., Lattice structure and convergence of a game of cards. Ann. Combin. 6 (2002) 327335.
[9] Goles, E., Morvan, M. and Phan, H.D.. Sandpiles and order structure of integer partitions. Discrete Appl. Math. 117 (2002) 5164.
[10] Goles, E., Morvan, M. and Phan, H.D., The structure of linear chip firing game and related models. Theoret. Comput. Sci. 270 (2002) 827841.
[11] Goles, É. and Latapy, M., Clémence Magnien, Michel Morvan and Ha Duong Phan. Sandpile models and lattices : a comprehensive survey. Theoret. Comput. Sci. 322 (2004) 383407.
[12] Huang, S.-T.. Leader election in uniform rings. ACM Trans. Program. Lang. Syst. 15 (1993) 563573.
[13] Karmakar, R. and Manna, S.S., Particle hole symmetry in a sandpile model, J. Stat. Mech. 2005 (2005) L01002.
[14] Latapy, M. and Phan, H.D., The lattice structure of chip firing games. Physica D 115 (2001) 6982.
[15] Phan, H.D., Two sided sand piles model and unimodal sequences. RAIRO – Theor. Inf. Appl. 42 (2008) 631646.


Related content

Powered by UNSILO

Signed Chip Firing Games and symmetric Sandpile Models on the cycles

  • Robert Cori (a1), Thi Ha Duong Phan (a2) and Thi Thu Huong Tran (a2)


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed.