Subgraphs in random graphs with specified degrees and forbidden edges

TL;DR

This paper refines probability bounds for subgraph occurrences in random graphs with specified degrees, allowing for broader degree sequences and complex edge constraints, including bipartite graphs and large forbidden subgraphs.

Contribution

It extends existing bounds on subgraph probabilities in random graphs to wider degree sequences and complex edge restrictions, using a more precise edge switching method.

Findings

Improved bounds for subgraph probabilities in random graphs.

Extended analysis to bipartite graphs with given degree sequences.

Permitted vertices with linear degree under certain conditions.

Abstract

Let $G$ be a uniformly chosen simple (labelled) random graph with given degree sequence $\boldsymbol{d}$ and let $X,Y,L$ be edge-disjoint graphs on the same vertex set as $G$. We investigate the probability that $X \subseteq G$ and that $G \cap Y = \emptyset$ both conditioned on the event $G \cap L = \emptyset$. We improve upon known bounds of these probabilities and extend them to a wider range of degree sequences through a more precise edge switching argument. Notably, a few vertices of linear degree are permitted provided that the subgraph $X$ does not have an edge incident with them. Further, the graph $L$ is permitted to contain many edges (we provide an example where $L$ is a spanning $r$-regular subgraph with $r = o(n)$). We provide the same analysis when $G$ is a simple (labelled) bipartite random graph with a given degree sequence $(\boldsymbol{s},\boldsymbol{t})$. Our work extends the results of Gao and Ohapkin (2023) and McKay (1981, 2010).