Subgraphs in random graphs with specified degrees and forbidden edges
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| Format: | Preprint |
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2025
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| _version_ | 1866914118265470976 |
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| author | Larkin, John McKay, Brendan D. Tian, Fang |
| author_facet | Larkin, John McKay, Brendan D. Tian, Fang |
| contents | 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). |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2510_24276 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Subgraphs in random graphs with specified degrees and forbidden edges Larkin, John McKay, Brendan D. Tian, Fang Combinatorics 05C80, 05C30 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). |
| title | Subgraphs in random graphs with specified degrees and forbidden edges |
| topic | Combinatorics 05C80, 05C30 |
| url | https://arxiv.org/abs/2510.24276 |