Subgraph probability of random graphs with specified degrees and applications to chromatic number and connectivity

Understand

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.