Graph Pebbling

2013 ◽  
pp. 1428-1449 ◽  
Author(s):  
Glenn Hurlbert
Keyword(s):  
Graph Pebbling

scholarly journals The Complexity of Graph Pebbling

2006 ◽  
Vol 20 (3) ◽  
pp. 769-798 ◽  
Author(s):  
Kevin Milans ◽  
Bryan Clark
Keyword(s):  
Graph Pebbling

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

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].

scholarly journals On Pebbling Graphs by Their Blocks

Integers ◽  
2009 ◽  
Vol 9 (4) ◽  
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

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

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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