Subgraphs in random graphs with specified degrees and forbidden edges

TL;DR

This work analyzes the probability that a fixed subgraph $X$ is contained in a uniformly random graph with a prescribed degree sequence, and the probability that a forbidden subgraph $Y$ avoids appearing, all under conditioning that another edge-set $L$ is absent. The authors extend the classic switchings method by introducing a degree-sum framework and a family of error-controlled functions $f$ and $g$, enabling sharper asymptotics for a wider class of degree sequences, including those with vertices of linear degree, and even when $L$ can be sizeable. They develop parallel results for bipartite graphs and provide comprehensive proofs via forward and backward switchings, culminating in single-edge, multiple-edge, and forbidden-edge theorems for both generic and bipartite graphs. The paper also offers concrete examples and calculations to demonstrate the practical reach of the bounds, illustrating improvements over prior results by Gao–Ohapkin and McKay. Overall, the results broaden the applicability of subgraph-probability estimates in random graphs with prescribed degrees, with explicit asymptotics under relatively mild degree-variation 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).

Figures (12)

• Figure 1: A 2-switching between $\mathcal{G}_{\boldsymbol{d}}(uv,M)$ and $\mathcal{G}_{\boldsymbol{d}}(\boldsymbol{\emptyset},M+uv)$.
• Figure 2: Bad cases where Condition 2 is not satisfied
• Figure 3: A 3-switching between $\mathcal{G}_{\boldsymbol{d}}(uv,M)$ and $\mathcal{G}_{\boldsymbol{d}}(\boldsymbol{\emptyset},M+uv)$
• Figure 4: Terrible cases, satisfying Condition 4 but not Condition 5
• Figure 5: Some bad cases, satisfying Conditions 4 and 5 but not Condition 6

Theorems & Definitions (60)

• Remark 1.1
• Theorem 2.1: Gao and Ohapkin, Theorem 4 gao22
• Theorem 2.2: Theorem 3.5 mckay81 - Simplification
• Definition 3.1: Lazy Degree Sum Function
• Definition 3.2: Degree Sum Function
• Lemma 3.3: Properties of the Degree Sum Function
• Definition 3.4: $\alpha$-value
• Definition 3.5: Uniform Bound on $\alpha$-Value
• Definition 4.1
• Lemma 4.2: Single Edge Probability