126 research outputs found
Optimal Pebbling in Products of Graphs
We prove a generalization of Graham's Conjecture for optimal pebbling with
arbitrary sets of target distributions. We provide bounds on optimal pebbling
numbers of products of complete graphs and explicitly find optimal -pebbling
numbers for specific such products. We obtain bounds on optimal pebbling
numbers of powers of the cycle . Finally, we present explicit
distributions which provide asymptotic bounds on optimal pebbling numbers of
hypercubes.Comment: 28 pages, 1 figur
Critical Pebbling Numbers of Graphs
We define three new pebbling parameters of a connected graph , the -,
-, and -critical pebbling numbers. Together with the pebbling number, the
optimal pebbling number, the number of vertices and the diameter of the
graph, this yields 7 graph parameters. We determine the relationships between
these parameters. We investigate properties of the -critical pebbling
number, and distinguish between greedy graphs, thrifty graphs, and graphs for
which the -critical pebbling number is .Comment: 26 page
Modified Linear Programming and Class 0 Bounds for Graph Pebbling
Given a configuration of pebbles on the vertices of a connected graph , a
\emph{pebbling move} removes two pebbles from some vertex and places one pebble
on an adjacent vertex. The \emph{pebbling number} of a graph is the
smallest integer such that for each vertex and each configuration of
pebbles on there is a sequence of pebbling moves that places at least
one pebble on .
First, we improve on results of Hurlbert, who introduced a linear
optimization technique for graph pebbling. In particular, we use a different
set of weight functions, based on graphs more general than trees. We apply this
new idea to some graphs from Hurlbert's paper to give improved bounds on their
pebbling numbers.
Second, we investigate the structure of Class 0 graphs with few edges. We
show that every -vertex Class 0 graph has at least
edges. This disproves a conjecture of Blasiak et al. For diameter 2 graphs, we
strengthen this lower bound to , which is best possible. Further, we
characterize the graphs where the bound holds with equality and extend the
argument to obtain an identical bound for diameter 2 graphs with no cut-vertex.Comment: 19 pages, 8 figure
Constructions for the optimal pebbling of grids
In [C. Xue, C. Yerger: Optimal Pebbling on Grids, Graphs and Combinatorics]
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 . First we present such a distribution with
covering ratio , disproving the conjecture. The authors in the above paper
also claim to prove that the covering ratio of any pebble distribution is at
most . 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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