Automating Weight Function Generation in Graph Pebbling

Graph pebbling is a combinatorial game played on an undirected graph with an initial configuration of pebbles. A pebbling move consists of removing two pebbles from one vertex and placing one pebble on an adjacent vertex. The pebbling number of a graph is the smallest number of pebbles necessary such that, given any initial configuration of pebbles, at least one pebble can be moved to a specified root vertex. Recent lines of inquiry apply computational techniques to pebbling bound generation and improvement. Along these lines, we present a computational framework that produces a set of tree strategy weight functions that are capable of proving pebbling number upper bounds on a connected graph. Our mixed-integer linear programming approach automates the generation of large sets of such functions and provides verifiable certificates of pebbling number upper bounds. The framework is capable of producing verifiable pebbling bounds on any connected graph, regardless of its structure or pebbling properties. We apply the model to the 4th weak Bruhat to prove $\pi(B_4) \leq 66$ and to the Lemke square graph to produce a set of certificates that verify $\pi(L x L) \leq 96$

Graph Pebbling: Doppelgangers and Lemke Graphs

Graph pebbling is a mathematical game played on a connected graph. A configuration places a nonnegative number of pebbles on each vertex. A move between a pair of adjacent vertices removes two pebbles from one vertex and places one pebble on the other. A configuration is solvable if a sequence of pebbling moves can be made to place a pebble on any given vertex. The pebbling number of a graph is the minimum number of pebbles so that any configuration is solvable. A graph satisfies the two-pebbling property if any configuration of more than twice the pebbling number minus the number of vertices with pebbles allows for placing two pebbles on any vertex after applying a sequence of pebbling moves. If a graph does not have the two-pebbling property, it is a Lemke graph. We say that two vertices are doppelgangers if they are adjacent to the same vertices. By adding vertices to a graph that are doppelgangers of existing vertices, the pebbling number does not decrease and does not increases more than the number of added vertices. By adding any number of doppelgangers to previous Lemke graphs in a particular manner, we are able to construct another graph that is also a Lemke graph. Additionally, we have created new algorithms to determine solvabilty. Using this along with four nondeterministic algorithms, we have been able to find all Lemke graphs with up to nine vertices