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Cover Pebbling

Cover Pebbling. Victor M. Moreno. California State University Channel Islands. Cycles and Graham’s Conjecture. Advisor: Dr. Cynthia Wyels. Sponsored by the Mathematical Association of America’s REU in Mathematics at California Lutheran University; funded by NSA and NSF. Definitions.

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Cover Pebbling

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  1. Cover Pebbling Victor M. Moreno California State University Channel Islands Cycles and Graham’s Conjecture Advisor: Dr. Cynthia Wyels Sponsored by the Mathematical Association of America’s REU in Mathematics at California Lutheran University; funded by NSA and NSF.

  2. Definitions • Distance • Diameter • Pebbling Move • Cover Pebbling Number • Support (G) • Simple Configuration

  3. u v Distance Distance, dist(u,v) is the length of a shortest path in G between u and v.

  4. Diameter Diameter, d(G)is the longest distance in a graph G u v

  5. Pebbling Move Pebbling Move is defined as removing two pebbles from a vertex and subsequently placing one pebble on an adjacent vertex. 1 2 1 4

  6. Cover Pebbling Number • Cover Pebbling Number of a graph G, , is the minimum number of pebbles needed to place a pebble on every vertex of Gsimultaneously regardless of initial configuration. 3 1 1 1 9 3 1 4 1 9 2 1 15 1 1 7 3 1 1

  7. Support (G) • The Support of a configuration is the subset of vertices of the graph that have at least one pebble. u v 3 2

  8. Simple Configuration • Simple Configuration: we say we have a simple configuration when the support subset consists of one vertex. u 15

  9. Cover Pebbling for Paths v w

  10. Cover Pebbling for Complete Graphs

  11. Conjecture 1 There exists a Simple configuration for which . Simple Configuration Conjecture Which configurations are the largest? Where is the number of pebbles in a Simple Configuration. Simple configurations are largest for both Paths and Complete graphs.

  12. Cover Pebbling for Cycles What is its Cover Pebbling Number? Can we generalize for all ? Is there an easier way to find it?

  13. Cover Pebbling for Cycles .

  14. Cover Pebbling for Cycles . Two cases: odd and even.

  15. Cover Pebbling for Cycles Case one : n is odd

  16. Cover Pebbling for Cycles Case two : n is even

  17. Cover Pebbling for Cycles Case two : n is even

  18. Cover Pebbling for Cycles

  19. Graham’s Conjecture: For any two graphs G and H, Graham’s Conjecture Graham’s Conjecture (Cover): For any two graphs G and H,

  20. Graham’s Conjecture (Proof)

  21. 1 1 1 1 1 1 1 2 5 G H 2 2 2 2 2 2 3 3 3 3 3 3 3 4

  22. Graham’s Conjecture (Proof) continued…

  23. Graham’s Conjecture (Proof) continued…

  24. Graham’s Conjecture (Proof)

  25. Conjecture 1 There exists a Simple configuration for which . Open Question Where is the number of pebbles in a Simple Configuration?.

  26. References [1] Wyels, Cynthia; “Optimal Pebbling of Paths and Cycles” May 30, 2003, pg 6. [2] Sjöstrand, Jonas; “The Cover Pebbling Theorem, arXiv: math.CO/0410129 v1; October 6.

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