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| Format: | Preprint |
| Published: |
2026
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2605.25515 |
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| _version_ | 1866911715379118080 |
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| author | Korsky, Samuel |
| author_facet | Korsky, Samuel |
| contents | Korsky, Saffat and Aiylam introduced a growth constant $c(G)$ for integer-valued $h$-Lipschitz functions on a finite graph $G$ and proved that, for $G=G(n,d/n)$, \[
\frac{1}{2d}+O(d^{-2})\le \log c(G)\le
\frac{4\log^2 d}{d}+O(d^{-1}) \] with high probability. We sharpen the random-graph part of their result; as $n\to\infty$ and then $d\to\infty$, we prove \[
\log c(G)=\frac{π^2}{6d}+o(d^{-1}) \] with high probability. Additionally, we derive bounds on $\log c(Q_d)$ where $Q_d$ is the $d$-dimensional hypercube graph: \[
\frac{π^2}{6d}+o(d^{-1}) \le \log{c(Q_d)}\le
\left(\frac{3}{4} + o(1)\right)\frac{\log d}{d}. \] |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_25515 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Lipschitz Functions on Sparse Graphs II Korsky, Samuel Combinatorics Korsky, Saffat and Aiylam introduced a growth constant $c(G)$ for integer-valued $h$-Lipschitz functions on a finite graph $G$ and proved that, for $G=G(n,d/n)$, \[ \frac{1}{2d}+O(d^{-2})\le \log c(G)\le \frac{4\log^2 d}{d}+O(d^{-1}) \] with high probability. We sharpen the random-graph part of their result; as $n\to\infty$ and then $d\to\infty$, we prove \[ \log c(G)=\frac{π^2}{6d}+o(d^{-1}) \] with high probability. Additionally, we derive bounds on $\log c(Q_d)$ where $Q_d$ is the $d$-dimensional hypercube graph: \[ \frac{π^2}{6d}+o(d^{-1}) \le \log{c(Q_d)}\le \left(\frac{3}{4} + o(1)\right)\frac{\log d}{d}. \] |
| title | Lipschitz Functions on Sparse Graphs II |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2605.25515 |