Cover pebbling number of Comb, Friendship and Helm graphs

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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The cover pebbling number of graphs

Discrete Mathematics ◽

10.1016/j.disc.2005.03.009 ◽

2005 ◽

Vol 296 (1) ◽

pp. 15-23 ◽

Cited By ~ 18

Author(s):

Betsy Crull ◽

Tammy Cundiff ◽

Paul Feltman ◽

Glenn H. Hurlbert ◽

Lara Pudwell ◽

...

Keyword(s):

Pebbling Number

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The optimal t-pebbling number of a certain complete m-ary tree

Revista de la Real Academia de Ciencias Exactas Físicas y Naturales Serie A Matemáticas ◽

10.1007/s13398-018-0503-2 ◽

2018 ◽

Vol 113 (3) ◽

pp. 2889-2910

Author(s):

Chin-Lin Shiue ◽

Hung-Hsing Chiang ◽

Mu-Ming Wong ◽

H. M. Srivastava

Keyword(s):

Pebbling Number

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Herscovici’s conjecture on C2n × G

Discrete Mathematics Algorithms and Applications ◽

10.1142/s1793830920500718 ◽

2020 ◽

Vol 12 (05) ◽

pp. 2050071

Author(s):

A. Lourdusamy ◽

T. Mathivanan

Keyword(s):

Positive Integer ◽

Connected Graph ◽

Adjacent Vertex ◽

Connected Graphs ◽

Number Formula ◽

Pebbling Number

The [Formula: see text]-pebbling number, [Formula: see text], of a connected graph [Formula: see text], is the smallest positive integer such that from every placement of [Formula: see text] pebbles, [Formula: see text] pebbles can be moved to any specified target vertex by a sequence of pebbling moves, each move taking two pebbles off a vertex and placing one on an adjacent vertex. A graph [Formula: see text] satisfies the [Formula: see text]-pebbling property if [Formula: see text] pebbles can be moved to any specified vertex when the total starting number of pebbles is [Formula: see text], where [Formula: see text] is the number of vertices with at least one pebble. We show that the cycle [Formula: see text] satisfies the [Formula: see text]-pebbling property. Herscovici conjectured that for any connected graphs [Formula: see text] and [Formula: see text], [Formula: see text]. We prove Herscovici’s conjecture is true, when [Formula: see text] is an even cycle and for variety of graphs [Formula: see text] which satisfy the [Formula: see text]-pebbling property.

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