Graph pebbling algorithms and Lemke 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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A Graph Pebbling Algorithm on Weighted Graphs

Journal of Graph Algorithms and Applications ◽

10.7155/jgaa.00205 ◽

2010 ◽

Vol 14 (2) ◽

pp. 221-244 ◽

Cited By ~ 6

Author(s):

Nandor Sieben

Keyword(s):

Weighted Graphs ◽

Graph Pebbling

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On Pebbling Graphs by Their Blocks

Integers ◽

10.1515/integ.2009.033 ◽

2009 ◽

Vol 9 (4) ◽

Cited By ~ 5

Author(s):

Dawn Curtis ◽

Taylor Hines ◽

Glenn Hurlbert ◽

Tatiana Moyer

Keyword(s):

Connected Graph ◽

Graph Pebbling

AbstractGraph pebbling is a game played on a connected graph

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Graph pebbling with many free pebbles can be difficult

Proceedings of the twelfth annual ACM symposium on Theory of computing - STOC '80 ◽

10.1145/800141.804681 ◽

1980 ◽

Cited By ~ 3

Author(s):

David A. Carlson ◽

John E. Savage

Keyword(s):

Graph Pebbling

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The $t$-Pebbling Number is Eventually Linear in $t$

The Electronic Journal of Combinatorics ◽

10.37236/640 ◽

2011 ◽

Vol 18 (1) ◽

Author(s):

Michael Hoffmann ◽

Jiří Matoušek ◽

Yoshio Okamoto ◽

Philipp Zumstein

Keyword(s):

Initial Distribution ◽

Adjacent Vertex ◽

Weighted Graphs ◽

Graph Pebbling ◽

Pebbling Number ◽

Integer Weight

In graph pebbling games, one considers a distribution of pebbles on the vertices of a graph, and a pebbling move consists of taking two pebbles off one vertex and placing one on an adjacent vertex. The $t$-pebbling number $\pi_t(G)$ of a graph $G$ is the smallest $m$ such that for every initial distribution of $m$ pebbles on $V(G)$ and every target vertex $x$ there exists a sequence of pebbling moves leading to a distibution with at least $t$ pebbles at $x$. Answering a question of Sieben, we show that for every graph $G$, $\pi_t(G)$ is eventually linear in $t$; that is, there are numbers $a,b,t_0$ such that $\pi_t(G)=at+b$ for all $t\ge t_0$. Our result is also valid for weighted graphs, where every edge $e=\{u,v\}$ has some integer weight $\omega(e)\ge 2$, and a pebbling move from $u$ to $v$ removes $\omega(e)$ pebbles at $u$ and adds one pebble to $v$.

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