Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Bootstrap percolation in Ore-type graphs   The $r$-neighbour bootstrap process describes an infection process on a
graph, where we start with a set of initially infected vertices and an
uninfected vertex becomes infected as soon as it has $r$ infected neighbours.
An inital set of infected vertices is called percolating if at the end of the
bootstrap process all vertices are infected. We give Ore-type conditions that
guarantee the existence of a small percolating set of size $l\leq 2r-2$ if the
number of vertices $n$ of our graph is sufficiently large: if $l\geq r$ and
satisfies $2r \geq l+2 \lfloor \sqrt{2(l-r)+0.25}+2.5 \rfloor-1$ then there
exists a percolating set of size $l$ for every graph in which any two
non-adjacent vertices $x$ and $y$ satisfy $deg(x)+deg(y) \geq
n+4r-2l-2\lfloor\sqrt{2(l-r)+0.25}+2.5 \rfloor-1$ and if $l$ is larger with
$l\leq 2r-2$ there exists a percolating set of size $l$ if $deg(x)+deg(y) \geq
n+2r-l-2$. Our results extend the work of Gunderson, who showed that a graph
with minimum degree $\lfloor n/2 \rfloor+r-3$ has a percolating set of size $r
\geq 4$. We also give bounds for arbitrarily large $l$ in the minimum degree
setting.

9
The categories of this paper are: nlin
4
The categories of this paper are: cond-mat
11
The categories of this paper are: hep-lat
22
The categories of this paper are: chao-dyn
29
The categories of this paper are: funct-an
21
The categories of this paper are: adap-org
16
The categories of this paper are: stat
14
The categories of this paper are: math-ph
8
The categories of this paper are: hep-ex
6
The categories of this paper are: astro-ph
35
The categories of this paper are: mtrl-th
26
The categories of this paper are: bayes-an
31
The categories of this paper are: ao-sci
25
The categories of this paper are: patt-sol
18
The categories of this paper are: econ
20
The categories of this paper are: acc-phys