64 research outputs found
Tournaments, 4-uniform hypergraphs, and an exact extremal result
We consider -uniform hypergraphs with the maximum number of hyperedges
subject to the condition that every set of vertices spans either or
exactly hyperedges and give a construction, using quadratic residues, for
an infinite family of such hypergraphs with the maximum number of hyperedges.
Baber has previously given an asymptotically best-possible result using random
tournaments. We give a connection between Baber's result and our construction
via Paley tournaments and investigate a `switching' operation on tournaments
that preserves hypergraphs arising from this construction.Comment: 23 pages, 6 figure
A sharp threshold for a modified bootstrap percolation with recovery
Bootstrap percolation is a type of cellular automaton on graphs, introduced
as a simple model of the dynamics of ferromagnetism. Vertices in a graph can be
in one of two states: `healthy' or `infected' and from an initial configuration
of states, healthy vertices become infected by local rules. While the usual
bootstrap processes are monotone in the sets of infected vertices, in this
paper, a modification is examined in which infected vertices can return to a
healthy state. Vertices are initially infected independently at random and the
central question is whether all vertices eventually become infected. The model
examined here is such a process on a square grid for which healthy vertices
with at least two infected neighbours become infected and infected vertices
with no infected neighbours become healthy. Sharp thresholds are given for the
critical probability of initial infections for all vertices eventually to
become infected.Comment: 45 page
Lower bounds for bootstrap percolation on Galton-Watson trees
Bootstrap percolation is a cellular automaton modelling the spread of an
`infection' on a graph. In this note, we prove a family of lower bounds on the
critical probability for -neighbour bootstrap percolation on Galton--Watson
trees in terms of moments of the offspring distributions. With this result we
confirm a conjecture of Bollob\'as, Gunderson, Holmgren, Janson and Przykucki.
We also show that these bounds are best possible up to positive constants not
depending on the offspring distribution.Comment: 7 page
Positive independence densities of finite rank countable hypergraphs are achieved by finite hypergraphs
The independence density of a finite hypergraph is the probability that a
subset of vertices, chosen uniformly at random contains no hyperedges.
Independence densities can be generalized to countable hypergraphs using
limits. We show that, in fact, every positive independence density of a
countably infinite hypergraph with hyperedges of bounded size is equal to the
independence density of some finite hypergraph whose hyperedges are no larger
than those in the infinite hypergraph. This answers a question of Bonato,
Brown, Kemkes, and Pra{\l}at about independence densities of graphs.
Furthermore, we show that for any , the set of independence densities of
hypergraphs with hyperedges of size at most is closed and contains no
infinite increasing sequences.Comment: To appear in the European Journal of Combinatorics, 12 page
The time of graph bootstrap percolation
Graph bootstrap percolation, introduced by Bollob\'as in 1968, is a cellular
automaton defined as follows. Given a "small" graph and a "large" graph , in consecutive steps we obtain from by
adding to it all new edges such that contains a new copy of
. We say that percolates if for some , we have .
For , the question about the size of the smallest percolating graphs
was independently answered by Alon, Frankl and Kalai in the 1980's. Recently,
Balogh, Bollob\'as and Morris considered graph bootstrap percolation for and studied the critical probability , for the event that
the graph percolates with high probability. In this paper, using the same
setup, we determine, up to a logarithmic factor, the critical probability for
percolation by time for all .Comment: 18 pages, 3 figure
Bounding the Number of Hyperedges in Friendship -Hypergraphs
For , an -uniform hypergraph is called a friendship
-hypergraph if every set of vertices has a unique 'friend' - that
is, there exists a unique vertex with the property that for each
subset of size , the set is a hyperedge.
We show that for , the number of hyperedges in a friendship
-hypergraph is at least , and we
characterise those hypergraphs which achieve this bound. This generalises a
result given by Li and van Rees in the case when .
We also obtain a new upper bound on the number of hyperedges in a friendship
-hypergraph, which improves on a known bound given by Li, van Rees, Seo and
Singhi when .Comment: 14 page
Limited packings of closed neighbourhoods in graphs
The k-limited packing number, , of a graph , introduced by
Gallant, Gunther, Hartnell, and Rall, is the maximum cardinality of a set
of vertices of such that every vertex of has at most elements of
in its closed neighbourhood. The main aim in this paper is to prove the
best-possible result that if is a cubic graph, then , improving the previous lower bound given by Gallant, \emph{et al.}
In addition, we construct an infinite family of graphs to show that lower
bounds given by Gagarin and Zverovich are asymptotically best-possible, up to a
constant factor, when is fixed and tends to infinity. For
tending to infinity and tending to infinity sufficiently
quickly, we give an asymptotically best-possible lower bound for ,
improving previous bounds
Random Geometric Graphs and Isometries of Normed Spaces
Given a countable dense subset of a finite-dimensional normed space ,
and , we form a random graph on by joining, independently and with
probability , each pair of points at distance less than . We say that
is `Rado' if any two such random graphs are (almost surely) isomorphic.
Bonato and Janssen showed that in almost all are Rado. Our
main aim in this paper is to show that is the unique normed space
with this property: indeed, in every other space almost all sets are
non-Rado. We also determine which spaces admit some Rado set: this turns out to
be the spaces that have an direct summand. These results answer
questions of Bonato and Janssen.
A key role is played by the determination of which finite-dimensional normed
spaces have the property that every bijective step-isometry (meaning that the
integer part of distances is preserved) is in fact an isometry. This result may
be of independent interest
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