{"paper":{"title":"Lipschitz Functions on Sparse Graphs","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Dhroova Aiylam, Samuel Korsky, Tahsin Saffat","submitted_at":"2024-01-14T07:43:40Z","abstract_excerpt":"In this work we attempt to count the number of integer-valued $h$-Lipschitz functions (functions that change by at most $h$ along edges) on two classes of sparse graphs; grid graphs $L_{m,n}$, and sparse random graphs $G(n,d/n)$. We find that for all $n$-vertex graphs $G$ with $k$ connected components, the number of such functions grows as $(ch)^{n - k}$ for some $1 \\le c \\le 2$. In particular, letting $\\alpha \\approx 1.16234$ be the largest solution to $\\tan{(1/x)} = x$, we prove that as $n \\to \\infty$ $$ c = \\alpha\\sqrt{2} \\approx 1.6438\\ \\ \\text{when}\\ \\ G = L_{2,n} $$ and $$ 1.351 \\approx "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2401.07223","kind":"arxiv","version":2},"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/2401.07223/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"}