{"paper":{"title":"The $\\Delta$ property: a bridge between split graphs and Number Theory","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.NT"],"primary_cat":"math.CO","authors_text":"Victor N. Schv\\\"ollner","submitted_at":"2026-05-24T21:34:09Z","abstract_excerpt":"For a split graph $S$, the combinatorics of 2-switches on $S$ is faithfully encoded by the factor graph $\\Phi(S)$, a multigraph whose induced cycles have length at most $4$. In this paper we address the following question: for which $n \\in \\mathbb{N}$ is there a split graph $S$ whose factor graph contains an $n$-simple triangle, that is, a triangle all of whose edges have multiplicity $n$? We show that the answer is governed by a purely arithmetic condition, the $\\Delta$ property, relating the differences and sums of complementary divisors of $n$, and thereby establish a two-way bridge between"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2605.25264","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/2605.25264/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"}