A remark on the independence number of sparse random Cayley sum graphs
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866909524045070336 |
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| author | Nenadov, Rajko |
| author_facet | Nenadov, Rajko |
| contents | The Cayley sum graph $Γ_S$ of a set $S \subseteq \mathbb{Z}_n$ is defined on the vertex set $\mathbb{Z}_n$, with an edge between distinct $x, y \in \mathbb{Z}_n$ if $x + y \in S$. Campos, Dahia, and Marciano have recently shown that if $S$ is formed by taking each element in $\mathbb{Z}_n$ independently with probability $p$, for $p > (\log n)^{-1/80}$, then with high probability the largest independent set in $Γ_S$ is of size $$
(2 + o(1)) \log_{1/(1-p)}(n). $$ This extends a result of Green and Morris, who considered the case $p = 1/2$, and asymptotically matches the independence number of the binomial random graph $G(n,p)$. We improve the range of $p$ for which this holds to $p > (\log n)^{-1/3 + o(1)}$. The heavy lifting has been done by Campos, Dahia, and Marciano, and we show that their key lemma can be used a bit more efficiently. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_02100 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A remark on the independence number of sparse random Cayley sum graphs Nenadov, Rajko Combinatorics The Cayley sum graph $Γ_S$ of a set $S \subseteq \mathbb{Z}_n$ is defined on the vertex set $\mathbb{Z}_n$, with an edge between distinct $x, y \in \mathbb{Z}_n$ if $x + y \in S$. Campos, Dahia, and Marciano have recently shown that if $S$ is formed by taking each element in $\mathbb{Z}_n$ independently with probability $p$, for $p > (\log n)^{-1/80}$, then with high probability the largest independent set in $Γ_S$ is of size $$ (2 + o(1)) \log_{1/(1-p)}(n). $$ This extends a result of Green and Morris, who considered the case $p = 1/2$, and asymptotically matches the independence number of the binomial random graph $G(n,p)$. We improve the range of $p$ for which this holds to $p > (\log n)^{-1/3 + o(1)}$. The heavy lifting has been done by Campos, Dahia, and Marciano, and we show that their key lemma can be used a bit more efficiently. |
| title | A remark on the independence number of sparse random Cayley sum graphs |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2503.02100 |