{"paper":{"title":"On $2$-connected graphs avoiding cycles of length $0$ modulo $4$","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Boram Park, Hojin Chu, Homoon Ryu","submitted_at":"2025-07-17T05:32:45Z","abstract_excerpt":"For two integers $k$ and $\\ell$, an $(\\ell \\text{ mod }k)$-cycle means a cycle of length $m$ such that $m\\equiv \\ell\\pmod{k}$. In 1977, Bollob\\'{a}s proved a conjecture of Burr and Erd\\H{o}s by showing that if $\\ell$ is even or $k$ is odd, then every $n$-vertex graph containing no $(\\ell \\text{ mod }k)$-cycles has at most a linear number of edges in terms of $n$. Since then, determining the exact extremal bounds for graphs without $(\\ell \\text{ mod }k)$-cycles has emerged as an interesting question in extremal graph theory, though the exact values are known only for a few integers $\\ell$ and $"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2507.12798","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/2507.12798/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"}