{"paper":{"title":"Lipschitz Functions on Sparse Graphs II","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Samuel Korsky","submitted_at":"2026-05-25T07:19:02Z","abstract_excerpt":"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 $G=G(n,d/n)$, \\[\n  \\frac{1}{2d}+O(d^{-2})\\le \\log c(G)\\le\n  \\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 \\[\n  \\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: \\[\n  \\frac{\\pi^2}{6d}+o(d^{-1}) \\le \\log{c(Q_d)}\\le\n  \\left(\\frac{3}{4"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2605.25515","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2605.25515/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}