General graph pebbling

Journal of Graph Theory, 2008

Given a distribution of pebbles on the vertices of a graph G, a pebbling move takes two pebbles from one vertex and puts one on a neighboring vertex. The pebbling number Π(G) is the least k such that for every distribution of k pebbles and every vertex r, a pebble can be moved to r. The optimal pebbling number Π OP T (G) is the least k such that some distribution of k pebbles permits reaching each vertex.

1999

We survey results on the pebbling numbers of graphs as well as their historical connection with a number-theoretic question of Erdős and Lemke. We also present new results on two probabilistic pebbling considerations, first the random graph threshold for the property that the pebbling number of a graph equals its number of vertices, and second the pebbling threshold function for various natural graph sequences. Finally, we relate the question of the existence of pebbling thresholds to a strengthening of the normal property of posets, and show that the multiset lattice is not supernormal.

2011

Graph pebbling is a network model for studying whether or not a given supply of discrete pebbles can satisfy a given demand via pebbling moves. A pebbling move across an edge of a graph takes two pebbles from one endpoint and places one pebble at the other endpoint; the other pebble is lost in transit as a toll. It has been shown that deciding whether a supply can meet a demand on a graph is NP-complete. The pebbling number of a graph is the smallest t such that every supply of t pebbles can satisfy every demand of one pebble. Deciding if the pebbling number is at most k is \Pi_2^P-complete. In this paper we develop a tool, called the Weight Function Lemma, for computing upper bounds and sometimes exact values for pebbling numbers with the assistance of linear optimization. With this tool we are able to calculate the pebbling numbers of much larger graphs than in previous algorithms, and much more quickly as well. We also obtain results for many families of graphs, in many cases by hand, with much simpler and remarkably shorter proofs than given in previously existing arguments (certificates typically of size at most the number of vertices times the maximum degree), especially for highly symmetric graphs. Here we apply the Weight Function Lemma to several specific graphs, including the Petersen, Lemke, 4th weak Bruhat, Lemke squared, and two random graphs, as well as to a number of infinite families of graphs, such as trees, cycles, graph powers of cycles, cubes, and some generalized Petersen and Coxeter graphs. This partly answers a question of Pachter, et al., by computing the pebbling exponent of cycles to within an asymptotically small range. It is conceivable that this method yields an approximation algorithm for graph pebbling.

Integers, 2000

Graph pebbling is a game played on a connected graph G. A player purchases pebbles at a dollar a piece and hands them to an adversary who distributes them among the vertices of G (called a configuration) and chooses a target vertex r. The player may make a pebbling move by taking two pebbles off of one vertex and moving one of them to a neighboring vertex. The player wins the game if he can move k pebbles to r. The value of the game (G, k), called the k-pebbling number of G and denoted π k (G), is the minimum cost to the player to guarantee a win. That is, it is the smallest positive integer m of pebbles so that, from every configuration of size m, one can move k pebbles to any target. In this paper, we use the block structure of graphs to investigate pebbling numbers, and we present the exact pebbling number of the graphs whose blocks are complete. We also provide an upper bound for the k-pebbling number of diameter-two graphs, which can be the basis for further investigation into the pebbling numbers of graphs with blocks that have diameter at most two.

Journal of Combinatorial Optimization, 2016

Given a configuration of pebbles on the vertices of a connected graph G, a pebbling move removes two pebbles from some vertex and places one pebble on an adjacent vertex. The pebbling number of a graph G is the smallest integer k such that for each vertex v and each configuration of k pebbles on G there is a sequence of pebbling moves that places at least one pebble on v. First, we improve on results of Hurlbert, who introduced a linear optimization technique for graph pebbling. In particular, we use a different set of weight functions, based on graphs more general than trees. We apply this new idea to some graphs from Hurlbert's paper to give improved bounds on their pebbling numbers. Second, we investigate the structure of Class 0 graphs with few edges. We show that every n-vertex Class 0 graph has at least 5 3 n− 11 3 edges. This disproves a conjecture of Blasiak et al. For diameter 2 graphs, we strengthen this lower bound to 2n − 5, which is best possible. Further, we characterize the graphs where the bound holds with equality and extend the argument to obtain an identical bound for diameter 2 graphs with no cut-vertex.

2004

A configuration of pebbles on the vertices of a graph is solvable if one can place a pebble on any given root vertex via a sequence of pebbling steps. The pebbling number of a graph G is the minimum number pi(G) so that every configuration of pi(G) pebbles is solvable. A graph is Class 0 if its pebbling number equals its number of vertices. A function is a pebbling threshold for a sequence of graphs if a randomly chosen configuration of asymptotically more pebbles is almost surely solvable, while one of asymptotically fewer pebbles is almost surely not. Here we prove that graphs on n>=9 vertices having minimum degree at least floor(n/2) are Class 0, as are bipartite graphs with m>=336 vertices in each part having minimum degree at least floor(m/2)+1. Both bounds are best possible. In addition, we prove that the pebbling threshold of graphs with minimum degree d, with sqrt{n} << d, is O(n^{3/2}/d), which is tight when d is proportional to n.

2008

Let G be a connected graph with the vertex set V and the edge set E, where |V | = n and |E| = m. Define a pebbling configuration as a function C : V → Z+ where C(v) represents the number of pebbles placed on vertex v. For any vertex v such that C(v) ≥ 2 a pebbling step consists of placing a pebble on one of the vertices adjacent to v and discarding two pebbles from v. A configuration is called r-solvable if there is a sequence of pebbling steps that places at least one pebble on vertex r. Any such sequence is called an r-solution. A configuration is called solvable if it is r-solvable for any r ∈ V. We call an r-solution minimal if it contains the smallest number of pebbling steps. The pebbling number of a graph G, denoted π(G), is the minimum number of pebbles such that the configuration is solvable no matter how the pebbles are distriibuted on the vertices. For any two vertices u, v ∈ V , the distance between u and v (denoted d(u, v)) is the the number of edges on the shortest pat...

2011

We expand the theory of pebbling to graphs with weighted edges. In a weighted pebbling game, one player distributes a set amount of weight on the edges of a graph and his opponent chooses a target vertex and places a configuration of pebbles on the vertices. Player one wins if, through a series of pebbling moves, he can move at

Discrete Mathematics, 2008

Consider a configuration of pebbles distributed on the vertices of a connected graph of order n. A pebbling step consists of removing two pebbles from a given vertex and placing one pebble on an adjacent vertex. A distribution of pebbles on a graph is called solvable if it is possible to place a pebble on any given vertex using a sequence of pebbling steps. The pebbling number of a graph, denoted f (G), is the minimal number of pebbles such that every configuration of f (G) pebbles on G is solvable. We derive several general upper bounds on the pebbling number, improving previous results.

2005

The subject of graph pebbling has seen dramatic growth recently, both in the number of publications and in the breadth of variations and applications. Here we update the reader on the many developments that have occurred since the original Survey of Graph Pebbling in 1999.