Lipschitz Functions on Sparse Graphs II

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: $$\frac{\pi^2}{6d}+o(d^{-1}) \le \log{c(Q_d)}\le \left(\frac{3}{4} + o(1)\right)\frac{\log d}{d}.$$