Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Percolating sets in bootstrap percolation on the Hamming graphs   For any integer $r\geqslant0$, the $r$-neighbor bootstrap percolation on a
graph is an activation process of the vertices. The process starts with some
initially activated vertices and then, in each round, any inactive vertex with
at least $r$ active neighbors becomes activated. A set of initially activated
vertices leading to the activation of all vertices is said to be a percolating
set. Denote the minimum size of a percolating set in the $r$-neighbor bootstrap
percolation process on a graph $G$ by $m(G, r)$. In this paper, we present
upper and lower bounds on $m(K_n^d, r)$, where $K_n^d$ is the Cartesian product
of $d$ copies of the complete graph $K_n$ which is referred as the Hamming
graph. Among other results, we show that $m(K_n^d, r)=\frac{1+o(1)}{(d+1)!}r^d$
when both $r$ and $d$ go to infinity with $r<n$ and $d=o(\!\sqrt{r})$.

11
The categories of this paper are: hep-lat
6
The categories of this paper are: astro-ph
17
The categories of this paper are: q-fin
27
The categories of this paper are: comp-gas
9
The categories of this paper are: nlin
18
The categories of this paper are: econ
3
The categories of this paper are: physics
37
The categories of this paper are: supr-con
20
The categories of this paper are: acc-phys
0
The categories of this paper are: hep-ph
26
The categories of this paper are: bayes-an
7
The categories of this paper are: hep-th
15
The categories of this paper are: nucl-ex
13
The categories of this paper are: quant-ph
21
The categories of this paper are: adap-org
30
The categories of this paper are: q-alg