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| Format: | Preprint |
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2025
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| Online-Zugang: | https://arxiv.org/abs/2507.01138 |
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| _version_ | 1866913969153769472 |
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| author | Alon, Noga Benjamini, Itai Zakharov, Georgii Zhukovskii, Maksim |
| author_facet | Alon, Noga Benjamini, Itai Zakharov, Georgii Zhukovskii, Maksim |
| contents | Let $G$ be a graph on $n$ vertices and $(H,+)$ be an abelian group. What is the minimum size ${\sf S}_H(G)$ of the set of all sums $A(u)+A(v)$ over all injections $A:V(G)\to H$? In 2012, the first author, Angel, the second author, and Lubetzky proved that, for expander graphs and $H=\mathbb{Z}$, this minimum is at least $Ω(\log n)$, and this bound is tight -- there exists a regular expander $G$ with ${\sf S}_{\mathbb{Z}}(G)=O(\log n)$. We prove that, for every constant $d\geq 3$, the random $d$-regular graph $\mathcal{G}_{n,d}$ has significantly larger sum-sets: with high probability, for every abelian group $H$, ${\sf S}_H(\mathcal{G}_{n,d})=Ω(n^{1-2/d})$. In particular, this proves that, for every $\varepsilon>0$, there exists a regular graph with $O(n)$ edges and with sum-sets of size at least $n^{1-\varepsilon}$, for all abelian groups.
The bound ${\sf S}_H(\mathcal{G}_{n,d})=Ω(n^{1-2/d})$ is tight up to a polylogarithmic factor: We show that, for every $3\leq d\leq \ln n/ \ln \ln n$, there exists an abelian group $H$ such that, for every graph $G$ on $n$ vertices with maximum degree at most $d$, ${\sf S}_H(G) \leq n^{1-2/d}(\log n)^{O(1)}$.
We also prove that, for $d\gg\ln^2 n$, with high probability, for every abelian group $H$, ${\sf S}_H(\mathcal{G}_{n,d})=n(1-o(1))$ and determine the second-order term, up to a polylogarithmic factor. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_01138 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Sums along the edges of bounded degree graphs Alon, Noga Benjamini, Itai Zakharov, Georgii Zhukovskii, Maksim Combinatorics Let $G$ be a graph on $n$ vertices and $(H,+)$ be an abelian group. What is the minimum size ${\sf S}_H(G)$ of the set of all sums $A(u)+A(v)$ over all injections $A:V(G)\to H$? In 2012, the first author, Angel, the second author, and Lubetzky proved that, for expander graphs and $H=\mathbb{Z}$, this minimum is at least $Ω(\log n)$, and this bound is tight -- there exists a regular expander $G$ with ${\sf S}_{\mathbb{Z}}(G)=O(\log n)$. We prove that, for every constant $d\geq 3$, the random $d$-regular graph $\mathcal{G}_{n,d}$ has significantly larger sum-sets: with high probability, for every abelian group $H$, ${\sf S}_H(\mathcal{G}_{n,d})=Ω(n^{1-2/d})$. In particular, this proves that, for every $\varepsilon>0$, there exists a regular graph with $O(n)$ edges and with sum-sets of size at least $n^{1-\varepsilon}$, for all abelian groups. The bound ${\sf S}_H(\mathcal{G}_{n,d})=Ω(n^{1-2/d})$ is tight up to a polylogarithmic factor: We show that, for every $3\leq d\leq \ln n/ \ln \ln n$, there exists an abelian group $H$ such that, for every graph $G$ on $n$ vertices with maximum degree at most $d$, ${\sf S}_H(G) \leq n^{1-2/d}(\log n)^{O(1)}$. We also prove that, for $d\gg\ln^2 n$, with high probability, for every abelian group $H$, ${\sf S}_H(\mathcal{G}_{n,d})=n(1-o(1))$ and determine the second-order term, up to a polylogarithmic factor. |
| title | Sums along the edges of bounded degree graphs |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2507.01138 |