Critical Pebbling Numbers of Graphs

With Dr. Joshua Laison and Erick J. Paul. Submitted to Discrete Math in September of 2004.

Abstract:

If some pebbles are distributed on the vertices of a graph, a pebbling step takes two pebbles from one vertex and replaces one at an adjacent vertex. A distribution D of pebbles is solvable if, starting from D, a pebble can be moved to any speciļ¬ed vertex by a sequence of pebbling steps. The pebbling number p(G) of a connected graph is the smallest number of pebbles such that every distribution with p(G) pebbles is solvable. If we restrict our attention to distributions which are minimally solvable or maximally unsolvable, we obtain three additional pebbling parameters on G, the r-, g-, and u-critical pebbling numbers of a connected graph. We investigate properties of the r-critical pebbling number and its relationship to the pebbling number.

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