50 research outputs found
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
Optimal pebbling and rubbling of graphs with given diameter
A pebbling move on a graph removes two pebbles from a vertex and adds one
pebble to an adjacent vertex. A vertex is reachable from a pebble distribution
if it is possible to move a pebble to that vertex using pebbling moves. The
optimal pebbling number is the smallest number needed to
guarantee a pebble distribution of pebbles from which any vertex is
reachable. A rubbling move is similar to a pebbling move, but it can remove the
two pebbles from two different vertex. The optimal rubbling number
is defined analogously to the optimal pebbling number.
In this paper we give lower bounds on both the optimal pebbling and rubbling
numbers by the distance domination number. With this bound we prove that
for each there is a graph with diameter such that
Optimal pebbling of grids
A pebbling move on a graph removes two pebbles at a vertex and adds
one pebble at an adjacent vertex. Rubbling is a version of pebbling
where an additional move is allowed. In this new move, one pebble
each is removed at vertices and adjacent to a vertex ,
and an extra pebble is added at vertex . A vertex is reachable
from a pebble distribution if it is possible to move a pebble to
that vertex using rubbling moves. The optimal pebbling (rubbling) number is
the smallest number needed to guarantee a pebble distribution of
pebbles from which any vertex is reachable using pebbling (rubbling) moves.
We determine the optimal rubbling number of ladders (), prisms
() and M\"oblus-ladders. We also give upper and lower
bounds for the optimal pebbling and rubbling numbers of large grids ()
Optimal Pebbling Number of the Square Grid
A pebbling move on a graph removes two pebbles from a vertex and adds one
pebble to an adjacent vertex. A vertex is reachable from a pebble distribution
if it is possible to move a pebble to that vertex using pebbling moves. The
optimal pebbling number is the smallest number m needed to
guarantee a pebble distribution of m pebbles from which any vertex is
reachable. The optimal pebbling number of the square grid graph was investigated in several papers. In this paper, we present a new method
using some recent ideas to give a lower bound on . We apply this
technique to prove that . Our
method also gives a new proof for
The optimal pebbling number of staircase graphs
Let G be a graph with a distribution of pebbles on its vertices. A pebbling move consists of removing two pebbles from one vertex and placing one pebble on an adjacent vertex. The optimal pebbling number of G is the smallest number of pebbles which can be placed on the vertices of G such that, for any vertex v of G, there is a sequence of pebbling moves resulting in at least one pebble on v. We determine the optimal pebbling number for several classes of induced subgraphs of the square grid, which we call staircase graphs. © 2018 Elsevier B.V
The history of degenerate (bipartite) extremal graph problems
This paper is a survey on Extremal Graph Theory, primarily focusing on the
case when one of the excluded graphs is bipartite. On one hand we give an
introduction to this field and also describe many important results, methods,
problems, and constructions.Comment: 97 pages, 11 figures, many problems. This is the preliminary version
of our survey presented in Erdos 100. In this version 2 only a citation was
complete