Graham’s pebbling conjecture on products of many cycles

Herscovici, David S.

doi:10.1016/j.disc.2007.12.045

Cited by 13 publications

(6 citation statements)

References 8 publications

(15 reference statements)

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“…The conjecture has been verified for many graphs; see [13] for the most recent work. However, as noted in [16], there is good reason to suspect that L L might be a counterexample to this conjecture, if one exists, where L is the Lemke graph of Figure 1.…”

Section: Conjecture 1 (Graham) Every Pair Of Graphs G and H Satisfy π...mentioning

confidence: 83%

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A linear optimization technique for graph pebbling

Hurlbert

2011

Preprint

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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 Π P 2 -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, 4 th 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. * Much of this work was carried out while vising the Centre de Recerca Matematica during the research program on Variational Analysis and Optimization, Fall 2010.

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Section: Conjecture 1 (Graham) Every Pair Of Graphs G and H Satisfy π...mentioning

confidence: 83%

“…Given our introductory comments and observations, we also tested some modest sized random graphs. Let R 15 denote the 15-vertex graph given by the adjacency list ( [2,4,5,6,12,13], [1,3,4,8,11,12,14], [2,4,6,7], [1,2,3,5,7,9,14], [1,4,6,8,11,15], [1,3,5,9,13,14], [3,4,11,15], [2,5,10,13,14,15], [4,6,10,11], [8,…”

Section: Random Graphsmentioning

confidence: 99%

A linear optimization technique for graph pebbling

Hurlbert

2011

Preprint

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“…While many useful results in support of this conjecture have been offered, the conjecture has yet to be proven for all graphs. The conjecture is verified when G and H are trees [6], cycles [2], a clique and a graph with the 2-pebbling property [1] and when G has the 2-pebbling property and H is a complete multi-partite graph [4].…”

Section: Conjecture 21 (Graham) For All Graphs G and H We Have π(G H)...mentioning

confidence: 91%

“…Conventional graph pebbling techniques employ topics in number theory [1,2,3,4], combinatorics [5,6] and graph theory [7,8,9]. Since the conceptualization of graph pebbling in 1989 [1], there have been many interesting and useful results presented through these theoretical mediums, particularly on small graphs and families of graphs.…”

Section: Introductionmentioning

confidence: 99%

Automating Weight Function Generation in Graph Pebbling

Flocco

Pulaj

Yerger

2022

Preprint

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“…By not specifying a target vertex, we postulate Conjectures 2.6 and 2.7. Conjecture 2.6 first appeared in [5], and Chung [1] attributed Conjecture 2.7 to Graham. CONJECTURE 2.6.…”

Section: Cartesian Products Definitionmentioning

confidence: 99%

Generalizations of Graham’s pebbling conjecture

Herscovici

Hester

Hurlbert

2012

Discrete Mathematics

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We investigate generalizations of pebbling numbers and of Graham's pebbling conjecture that π(G × H) ≤ π(G)π(H), where π(G) is the pebbling number of the graph G. We develop new machinery to attack the conjecture, which is now twenty years old. We show that certain conjectures imply others that initially appear stronger. We also find counterexamples that show that Sjöstrand's theorem on cover pebbling does not apply if we allow the cost of transferring a pebble from one vertex to an adjacent vertex to depend on the edge and we describe an alternate pebbling number for which Graham's conjecture is demonstrably false.

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Graham’s pebbling conjecture on products of many cycles

Cited by 13 publications

References 8 publications

A linear optimization technique for graph pebbling

A linear optimization technique for graph pebbling

Automating Weight Function Generation in Graph Pebbling

Generalizations of Graham’s pebbling conjecture

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