The optimal pebbling number of staircase graphs

Abstract Given a distribution of pebbles on the vertices of a connected graph G, a pebbling move on G consists of taking two pebbles off one vertex and placing one on an adjacent vertex. The optimal pebbling number of G, denoted by πopt(G), is the smallest number m such that for some distribution of m pebbles on G, one pebble can be moved to any vertex of G by a sequence of pebbling moves. Let Pk be the path on k vertices. Snevily defined the n–k spindle graph as follows: take n copies of Pk and two extra vertices x and y, and then join the left endpoint (respectively, the right endpoint) of each Pk to x (respectively, y), the resulting graph is denoted by S(n, k), and called the n–k spindle graph. In this paper, we determine the optimal pebbling number for spindle graphs.

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Optimal Pebbling Number of the Square Grid

Graphs and Combinatorics ◽

10.1007/s00373-020-02154-z ◽

2020 ◽

Vol 36 (3) ◽

pp. 803-829

Author(s):

Ervin Győri ◽

Gyula Y. Katona ◽

László F. Papp

Keyword(s):

Square Grid ◽

Pebbling Number ◽

Optimal Pebbling

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Cover Pebbling Number of Some Cycle Related Graphs

Journal of Computer and Mathematical Sciences ◽

10.29055/jcms/1127 ◽

2019 ◽

Vol 10 (6) ◽

pp. 1322-1331

Author(s):

Joice Punitha M ◽

Suganya A

Keyword(s):

Pebbling Number

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Distance restricted optimal pebbling in cycles

Discrete Applied Mathematics ◽

10.1016/j.dam.2019.10.017 ◽

2020 ◽

Vol 279 ◽

pp. 125-133

Author(s):

Chin-Lin Shiue

Keyword(s):

Optimal Pebbling

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Optimal Pebbling on Grids

Graphs and Combinatorics ◽

10.1007/s00373-015-1615-5 ◽

2015 ◽

Vol 32 (3) ◽

pp. 1229-1247 ◽

Cited By ~ 3

Author(s):

Chenxiao Xue ◽

Carl Yerger

Keyword(s):

Optimal Pebbling

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Graham’s pebbling conjecture holds for the product of a graph and a sufficiently large complete bipartite graph

Discrete Mathematics Algorithms and Applications ◽

10.1142/s179383091950068x ◽

2019 ◽

Vol 11 (06) ◽

pp. 1950068

Author(s):

Nopparat Pleanmani

Keyword(s):

Bipartite Graph ◽

Network Optimization ◽

Connected Graph ◽

Complete Bipartite Graph ◽

Fixed Number ◽

Arbitrary Vertex ◽

Graph Pebbling ◽

Vertex Set ◽

Graham’S Conjecture ◽

Pebbling Number

A graph pebbling is a network optimization model for the transmission of consumable resources. A pebbling move on a connected graph [Formula: see text] is the process of removing two pebbles from a vertex and placing one of them on an adjacent vertex after configuration of a fixed number of pebbles on the vertex set of [Formula: see text]. The pebbling number of [Formula: see text], denoted by [Formula: see text], is defined to be the least number of pebbles to guarantee that for any configuration of pebbles on [Formula: see text] and arbitrary vertex [Formula: see text], there is a sequence of pebbling movement that places at least one pebble on [Formula: see text]. For connected graphs [Formula: see text] and [Formula: see text], Graham’s conjecture asserted that [Formula: see text]. In this paper, we show that such conjecture holds when [Formula: see text] is a complete bipartite graph with sufficiently large order in terms of [Formula: see text] and the order of [Formula: see text].

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Constructions for the Optimal Pebbling of Grids

Periodica Polytechnica Electrical Engineering and Computer Science ◽

10.3311/ppee.9724 ◽

2017 ◽

Vol 61 (2) ◽

pp. 217 ◽

Cited By ~ 3

Author(s):

Ervin Győri ◽

Gyula Y. Katona ◽

László F. Papp

Keyword(s):

Square Grid ◽

Optimal Pebbling

In [6] the authors conjecture that if every vertex of an infinite square grid is reachable from a pebble distribution, then the covering ratio of this distribution is at most 3.25. First we present such a distribution with covering ratio 3.5, disproving the conjecture. The authors in the above paper also claim to prove that the covering ratio of any pebble distribution is at most 6.75. The proof contains some errors. We present a few interesting pebble distributions that this proof does not seem to cover and highlight some other difficulties of this topic.

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