A pebbling configuration on a simple, connected graph assigns a nonnegative integer to each vertex. A pebbling move removes two pebbles from one vertex and places one on an adjacent vertex. A vertex is reachable if a sequence of moves places a pebble on it, and a configuration is solvable if every vertex is reachable. The pebbling number π(G) is the minimum size of a solvable configuration on G. A graph satisfies the two-pebbling property if any configuration with more than 2π(G)−q pebbles, where q is the number of occupied vertices, can place two pebbles on any vertex; otherwise it is a Lemke graph. We present an algorithm to test reachability and solvability, describe methods for computing π(G) and the two-pebbling property, and determine all Lemke graphs on at most nine vertices, including many previously unknown examples.

Aaron Micah Green is currently teaching at Vassar College. He earned his PhD in Applied Math from RPI in August 2025. He previously worked as a software engineer for Epic Systems in Madison, WI.

For more information, please visit our website: Math Frontier Seminar Website.