73 research outputs found
Cover Pebbling Hypercubes
Given a graph G and a configuration C of pebbles on the vertices of G, a
pebbling step removes two pebbles from one vertex and places one pebble on an
adjacent vertex. The cover pebbling number g=g(G) is the minimum number so that
every configuration of g pebbles has the property that, after some sequence of
pebbling steps, every vertex has a pebble on it. We prove that the cover
pebbling number of the d-dimensional hypercube Q^d equals 3^d.Comment: 11 page
Pebbling in Dense Graphs
A configuration of pebbles on the vertices of a graph is solvable if one can
place a pebble on any given root vertex via a sequence of pebbling steps. The
pebbling number of a graph G is the minimum number pi(G) so that every
configuration of pi(G) pebbles is solvable. A graph is Class 0 if its pebbling
number equals its number of vertices. A function is a pebbling threshold for a
sequence of graphs if a randomly chosen configuration of asymptotically more
pebbles is almost surely solvable, while one of asymptotically fewer pebbles is
almost surely not. Here we prove that graphs on n>=9 vertices having minimum
degree at least floor(n/2) are Class 0, as are bipartite graphs with m>=336
vertices in each part having minimum degree at least floor(m/2)+1. Both bounds
are best possible. In addition, we prove that the pebbling threshold of graphs
with minimum degree d, with sqrt{n} << d, is O(n^{3/2}/d), which is tight when
d is proportional to n.Comment: 10 page
The Cover Pebbling Number of Graphs
A pebbling move on a graph consists of taking two pebbles off of one vertex
and placing one pebble on an adjacent vertex. In the traditional pebbling
problem we try to reach a specified vertex of the graph by a sequence of
pebbling moves. In this paper we investigate the case when every vertex of the
graph must end up with at least one pebble after a series of pebbling moves.
The cover pebbling number of a graph is the minimum number of pebbles such that
however the pebbles are initially placed on the vertices of the graph we can
eventually put a pebble on every vertex simultaneously. We find the cover
pebbling numbers of trees and some other graphs. We also consider the more
general problem where (possibly different) given numbers of pebbles are
required for the vertices.Comment: 12 pages. Submitted to Discrete Mathematic
Cover pebbling numbers and bounds for certain families of graphs
Given a configuration of pebbles on the vertices of a graph, a pebbling move
is defined by removing two pebbles from some vertex and placing one pebble on
an adjacent vertex. The cover pebbling number of a graph, gamma(G), is the
smallest number of pebbles such that through a sequence of pebbling moves, a
pebble can eventually be placed on every vertex simultaneously, no matter how
the pebbles are initially distributed. The cover pebbling number for complete
multipartite graphs and wheel graphs is determined. We also prove a sharp bound
for gamma(G) given the diameter and number of vertices of G.Comment: 10 pages, 1 figure, submitted to Discrete Mathematic
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