On the $H$-space of a random graph
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866912064953384960 |
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| author | Dubroff, Quentin Kahn, Jeff |
| author_facet | Dubroff, Quentin Kahn, Jeff |
| contents | The edge space $\mathcal{E}(G)$ of a graph $G$ is the vector space $\mathbb{F}_2^{E(G)}$ with members naturally identified with subgraphs of $G$, and the $H$-space is the subspace $\mathcal{C}_H(G)$ of $ \mathcal{E}(G)$ spanned by copies of the graph $H$. We are interested in when the random graph $G = G_{n,p}$ is likely to satisfy \[\mathcal{C}_H(G) = \mathcal{W}_H(G),\] where $\mathcal{W}_H(G)$ takes one of four natural values, depending on the value of $\mathcal{C}_H(K_n)$. We show that for strictly $2$-balanced $H$, w.h.p. the above equality holds whenever every edge of $G$ is in a copy of $H$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_06421 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the $H$-space of a random graph Dubroff, Quentin Kahn, Jeff Combinatorics Probability The edge space $\mathcal{E}(G)$ of a graph $G$ is the vector space $\mathbb{F}_2^{E(G)}$ with members naturally identified with subgraphs of $G$, and the $H$-space is the subspace $\mathcal{C}_H(G)$ of $ \mathcal{E}(G)$ spanned by copies of the graph $H$. We are interested in when the random graph $G = G_{n,p}$ is likely to satisfy \[\mathcal{C}_H(G) = \mathcal{W}_H(G),\] where $\mathcal{W}_H(G)$ takes one of four natural values, depending on the value of $\mathcal{C}_H(K_n)$. We show that for strictly $2$-balanced $H$, w.h.p. the above equality holds whenever every edge of $G$ is in a copy of $H$. |
| title | On the $H$-space of a random graph |
| topic | Combinatorics Probability |
| url | https://arxiv.org/abs/2410.06421 |