Subgraph probability of random graphs with specified degrees and applications to chromatic number and connectivity
Pu Gao, Yuval Ohapkin
TL;DR
This work develops probabilistic tools for random graphs with a fixed degree sequence by relating subgraph-edge probabilities in ${\mathcal{G}}(n,{f d})$ to the configuration model, under mild conditions such as $J({\bf d})=o(M)$. Through precise conditional and joint-edge probability bounds, the authors translate configuration-model analyses to the simple-graph setting, enabling streamlined proofs and improved results for sparse and dense regimes. They apply these tools to two central problems: the chromatic number, obtaining tight $\Theta(d/\ln d)$ bounds under weakened hypotheses, and the connectivity transition, fully characterizing the phase transition when $J({\bf d})=o(M)$ and providing sufficient conditions for connectivity in the unrestricted-degree case. The approach unifies and extends prior results (e.g., Frieze–Krivelevich–Smyth) and encompasses degree sequences with power-law behavior, regular sublinear sequences, and more, by leveraging robust switching arguments and sharp subgraph-probability bounds.
Abstract
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 $Δ^2=o(M)$ where $Δ$ 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 $Δ$ is unrestricted.