Random M-Lipschitz functions on graphs with slowly growing balls have range at least about M r/2, while on layered cycles C_{n,k} with k above γ M^2 log(Mn) the range is exactly M+1 with high probability.
Some intersection theorems for ordered sets and graphs
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Random Lipschitz functions on graphs with weak expansion
Random M-Lipschitz functions on graphs with slowly growing balls have range at least about M r/2, while on layered cycles C_{n,k} with k above γ M^2 log(Mn) the range is exactly M+1 with high probability.