Graham’s Conjecture, Lemke Graphs, and Algorithms
Dr. Charles Cusack and Dr. Airat Bekmetjev
Graph pebbling is a game on a connected graph G in which pebbles are placed on the vertices of G. A pebbling move consists of removing two pebbles from any vertex and adding one to an adjacent vertex. A configuration of pebbles is r-solvable if for a given target vertex r, there is a sequence of pebbling moves so that at least one pebble can be placed on r. The pebbling number of a graph G is the smallest integer π(G) such that any configuration that uses π(G) pebbles is r-solvable for any r V (G). A long standing conjecture in graph pebbling is Graham’s Conjecture. It states that given any two graphs G and H, π(G¨H) ≤ π(G)π(H), where G¨H is the Cartesian product of graphs. A graph G satisfies the two-pebbling property if two pebbles can be placed on any vertex v V (G) given any configuration of 2π(G)−q+1 pebbles, where q is the number of vertices that have at least one pebble. The only known graphs that do not satisfy the two-pebbling property are called Lemke graphs (Lk). We will show that Graham’s conjecture holds for such families as Lk¨Kn and several others.
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