Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Subgraph probability of random graphs with specified degrees and
  applications to chromatic number and connectivity   Given a graphical degree sequence ${\bf d}=(d_1,\ldots, d_n)$, let $G(n, {\bf
d})$ denote a uniformly random graph on vertex set $[n]$ where vertex $ i$ has
degree $d_i$ for every $1\le i\le n$. We give upper and lower bounds on the
joint probability of an arbitrary set of edges in $G(n,{\bf d})$. These upper
and lower bounds are approximately what one would get in the configuration
model, and thus the analysis in the configuration model can be translated
directly to $G(n,{\bf d})$, without conditioning on that the configuration
model produces a simple graph. Many existing results of $G(n,{\bf d})$ in the
literature can be significantly improved with simpler proofs, by applying this
new probabilistic tool. One example we give is about the chromatic number of
$G(n,{\bf d})$.
  In another application, we use these joint probabilities to study the
connectivity of $G(n,{\bf d})$. When $\Delta^2=o(M)$ where $\Delta$ is the
maximum component of ${\bf d}$, we fully characterise the connectivity phase
transition of $G(n,{\bf d})$. We also give sufficient conditions for $G(n,{\bf
d})$ being connected when $\Delta$ is unrestricted.

9
The categories of this paper are: nlin
8
The categories of this paper are: hep-ex
14
The categories of this paper are: math-ph
7
The categories of this paper are: hep-th
20
The categories of this paper are: acc-phys
25
The categories of this paper are: patt-sol
16
The categories of this paper are: stat
4
The categories of this paper are: cond-mat
37
The categories of this paper are: supr-con
11
The categories of this paper are: hep-lat
6
The categories of this paper are: astro-ph
23
The categories of this paper are: solv-int
17
The categories of this paper are: q-fin
21
The categories of this paper are: adap-org
15
The categories of this paper are: nucl-ex
24
The categories of this paper are: dg-ga