Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On random k-out sub-graphs of large graphs   We consider random sub-graphs of a fixed graph $G=(V,E)$ with large minimum
degree. We fix a positive integer $k$ and let $G_k$ be the random sub-graph
where each $v\in V$ independently chooses $k$ random neighbors, making $kn$
edges in all. When the minimum degree $\delta(G)\geq
(\frac12+\epsilon)n,\,n=|V|$ then $G_k$ is $k$-connected w.h.p. for $k=O(1)$;
Hamiltonian for $k$ sufficiently large. When $\delta(G) \geq m$, then $G_k$ has
a cycle of length $(1-\epsilon)m$ for $k\geq k_\epsilon$. By w.h.p. we mean
that the probability of non-occurrence can be bounded by a function $\phi(n)$
(or $\phi(m)$) where $\lim_{n\to\infty}\phi(n)=0$.

7
The categories of this paper are: hep-th
27
The categories of this paper are: comp-gas
16
The categories of this paper are: stat
37
The categories of this paper are: supr-con
25
The categories of this paper are: patt-sol
23
The categories of this paper are: solv-int
19
The categories of this paper are: eess
28
The categories of this paper are: alg-geom
14
The categories of this paper are: math-ph
29
The categories of this paper are: funct-an
35
The categories of this paper are: mtrl-th
18
The categories of this paper are: econ
13
The categories of this paper are: quant-ph
32
The categories of this paper are: atom-ph
3
The categories of this paper are: physics
9
The categories of this paper are: nlin