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arXiv · 2510.24276

Subgraphs in random graphs with specified degrees and forbidden edges

John Larkin·Brendan D. McKay·Fang Tian
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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).

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John Larkin, Brendan D. McKay, Fang Tian. 2025-10-28. Subgraphs in random graphs with specified degrees and forbidden edges. https://arxiv.org/abs/2510.24276

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