1. Cover Pebbling Cycles and Graham’s Conjecture Victor M. Moreno California State University Channel Islands 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 Distance Diameter Diameter Pebbling Move Pebbling Move Cover Pebbling Number Cover Pebbling Number Support (G) Support (G) Simple Configuration Simple Configuration

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

4. Diameter Diameter, d(G) is the longest distance in a graph G 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. Pebbling Move is defined as removing two pebbles from a vertex and subsequently placing one pebble on an adjacent vertex. 4 21 1

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 G simultaneously regardless of initial configuration. Cover Pebbling Number of a graph G,, is the minimum number of pebbles needed to place a pebble on every vertex of G simultaneously regardless of initial configuration. 9 3 3 1 1 1 1 92 1 1 4 15731111

7. Support (G) The Support of a configuration is the subset of vertices of the graph that have at least one pebble. The Support of a configuration is the subset of vertices of the graph that have at least one pebble. uv 32

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

9. Cover Pebbling for Paths vw

10. Cover Pebbling for Complete Graphs

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

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

13. Cover Pebbling for Cycles.

14. 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

18. Cover Pebbling for Cycles

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

20. Graham’s Conjecture (Proof)

21. G H 1 25 34 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3

22. Graham’s Conjecture (Proof) continued…

24. Graham’s Conjecture (Proof)

25. Open Question Conjecture 1 There exists a Simple configuration for which. 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.