Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Long paths and cycles in random subgraphs of graphs with large minimum
  degree   For a graph $G$ and $p\in [0,1]$, let $G_p$ arise from $G$ by deleting every
edge mutually independently with probability $1-p$. The random graph model
$(K_n)_p$ is certainly the most investigated random graph model and also known
as the $G(n,p)$-model. We show that several results concerning the length of
the longest path/cycle naturally translate to $G_p$ if $G$ is an arbitrary
graph of minimum degree at least $n-1$.
  For a constant $c$, we show that asymptotically almost surely the length of
the longest path is at least $(1-(1+\epsilon(c))ce^{-c})n$ for some function
$\epsilon(c)\to 0$ as $c\to \infty$, and the length of the longest cycle is a
least $(1-O(c^{- \frac{1}{5}}))n$. The first result is asymptotically
best-possible. This extents several known results on the length of the longest
path/cycle of a random graph in the $G(n,p)$-model.

23
The categories of this paper are: solv-int
7
The categories of this paper are: hep-th
14
The categories of this paper are: math-ph
36
The categories of this paper are: cmp-lg
27
The categories of this paper are: comp-gas
25
The categories of this paper are: patt-sol
10
The categories of this paper are: q-bio
6
The categories of this paper are: astro-ph
21
The categories of this paper are: adap-org
19
The categories of this paper are: eess
17
The categories of this paper are: q-fin
29
The categories of this paper are: funct-an
12
The categories of this paper are: nucl-th
8
The categories of this paper are: hep-ex
34
The categories of this paper are: chem-ph
11
The categories of this paper are: hep-lat