Abstract: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 \[
\log c(G)=\frac{\pi^2}{6d}+o(d^{-1}) \] with high probability. Additionally, we derive bounds on $\log c(Q_d)$ where $Q_d$ is the $d$-dimensional hypercube graph: \[
\frac{\pi^2}{6d}+o(d^{-1}) \le \log{c(Q_d)}\le
\left(\frac{3}{4} + o(1)\right)\frac{\log d}{d}. \]
| Subjects: | Combinatorics (math.CO) |
| Cite as: | arXiv:2605.25515 [math.CO] |
| (or arXiv:2605.25515v1 [math.CO] for this version) | |
| https://doi.org/10.48550/arXiv.2605.25515 arXiv-issued DOI via DataCite (pending registration) |
From: Samuel Korsky [view email]
[v1]
Mon, 25 May 2026 07:19:02 UTC (14 KB)
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