68 research outputs found
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
Thresholds for zero-sums with small cross numbers in abelian groups
For an additive group the sequence of
elements of is a zero-sum sequence if .
The cross number of is defined to be the sum , where
denotes the order of in . Call good if it contains a
zero-sum subsequence with cross number at most 1. In 1993, Geroldinger proved
that if is abelian then every length sequence of its
elements is good, generalizing a 1989 result of Lemke and Kleitman that had
proved an earlier conjecture of Erd\H{o}s and Lemke. In 1989 Chung re-proved
the Lemke and Kleitman result by applying a theorem of graph pebbling, and in
2005, Elledge and Hurlbert used graph pebbling to re-prove and generalize
Geroldinger's result. Here we use probabilistic theorems from graph pebbling to
derive a sharp threshold version of Geroldinger's theorem for abelian groups of
a certain form. Specifically, we prove that if are (not
necessarily distinct) primes and has the form then there is a function (which we specify
in Theorem 4) with the following property: if as
then the probability that is good in tends
to 1, while if then that probability tends to 0
Width and size of regular resolution proofs
This paper discusses the topic of the minimum width of a regular resolution
refutation of a set of clauses. The main result shows that there are examples
having small regular resolution refutations, for which any regular refutation
must contain a large clause. This forms a contrast with corresponding results
for general resolution refutations.Comment: The article was reformatted using the style file for Logical Methods
in Computer Scienc
Counterexamples to a Monotonicity Conjecture for the Threshold Pebbling Number
Graph pebbling considers the problem of transforming configurations of
discrete pebbles to certain target configurations on the vertices of a graph,
using the so-called pebbling move. This paper provides counterexamples to a
monotonicity conjecture stated by Hurlbert et al. concerning the pebbling
number compared to the pebbling threshold
The pebbling threshold of the square of cliques
AbstractGiven an initial configuration of pebbles on a graph, one can move pebbles in pairs along edges, at the cost of one of the pebbles moved, with the objective of reaching a specified target vertex. The pebbling number of a graph is the minimum number of pebbles so that every configuration of that many pebbles can reach any chosen target. The pebbling threshold of a sequence of graphs is roughly the number of pebbles so that almost every (resp. almost no) configuration of asymptotically more (resp. fewer) pebbles can reach any chosen target. In this paper we find the pebbling threshold of the sequence of squares of cliques, improving upon an earlier result of Boyle and verifying an important instance of a probabilistic version of Graham's product conjecture
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