15 research outputs found
Linear algebra and bootstrap percolation
In \HH-bootstrap percolation, a set A \subset V(\HH) of initially
'infected' vertices spreads by infecting vertices which are the only uninfected
vertex in an edge of the hypergraph \HH. A particular case of this is the
-bootstrap process, in which \HH encodes copies of in a graph . We
find the minimum size of a set that leads to complete infection when
and are powers of complete graphs and \HH encodes induced copies of
in . The proof uses linear algebra, a technique that is new in bootstrap
percolation, although standard in the study of weakly saturated graphs, which
are equivalent to (edge) -bootstrap percolation on a complete graph.Comment: 10 page
Exact Bounds for Some Hypergraph Saturation Problems
Let W_n(p,q) denote the minimum number of edges in an n x n bipartite graph G
on vertex sets X,Y that satisfies the following condition; one can add the
edges between X and Y that do not belong to G one after the other so that
whenever a new edge is added, a new copy of K_{p,q} is created. The problem of
bounding W_n(p,q), and its natural hypergraph generalization, was introduced by
Balogh, Bollob\'as, Morris and Riordan. Their main result, specialized to
graphs, used algebraic methods to determine W_n(1,q).
Our main results in this paper give exact bounds for W_n(p,q), its hypergraph
analogue, as well as for a new variant of Bollob\'as's Two Families theorem. In
particular, we completely determine W_n(p,q), showing that if 1 <= p <= q <= n
then
W_n(p,q) = n^2 - (n-p+1)^2 + (q-p)^2.
Our proof applies a reduction to a multi-partite version of the Two Families
theorem obtained by Alon. While the reduction is combinatorial, the main idea
behind it is algebraic
Line percolation
We study a new geometric bootstrap percolation model, line percolation, on
the -dimensional integer grid . In line percolation with infection
parameter , infection spreads from a subset of initially
infected lattice points as follows: if there exists an axis-parallel line
with or more infected lattice points on it, then every lattice point of
on gets infected, and we repeat this until the infection can no
longer spread. The elements of the set are usually chosen independently,
with some density , and the main question is to determine , the
density at which percolation (infection of the entire grid) becomes likely. In
this paper, we determine up to a multiplicative factor of
and up to a multiplicative constant as for
every fixed . We also determine the size of the minimal
percolating sets in all dimensions and for all values of the infection
parameter.Comment: 27 pages, Random Structures and Algorithm