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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"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"2605.25515","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2026-05-25T07:19:02Z","cross_cats_sorted":[],"title_canon_sha256":"1c948835e220fd8adac8b97b0a6849228191c33d0a5fc26840cb38b522d50a3a","abstract_canon_sha256":"77868ce5fb7c261804c5c94f7ad4bba3094d662cefb4e8b39842aa344e96bfcd"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-26T02:04:40.406336Z","signature_b64":"P5nni1/aTAwYO05p76pOy3TLAcPZWFUI5DVmJQVjIsHXd79M5obUZ9WUXl+dh73YJ4+agF5m465nVfE7bZ4QBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0a0b6bd0632c92f1d4fd24ff848be95c791a049c49ef78db8168f35ad8067da6","last_reissued_at":"2026-05-26T02:04:40.404775Z","signature_status":"signed_v1","first_computed_at":"2026-05-26T02:04:40.404775Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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. 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