{"paper":{"title":"On 2-connected graphs without cycles of length 1 modulo 3","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Binlong Li, Boram Park, Hojin Chu, Homoon Ryu, Yandong Bai","submitted_at":"2026-06-01T15:03:07Z","abstract_excerpt":"Burr and Erd\\H{o}s conjectured in 1976 that for all integers $k>\\ell\\geq 0$ such that $k\\mathbb{Z}+\\ell$ contains an even integer, every $n$-vertex graph without cycles of length $\\ell$ modulo $k$ has at most a linear number of edges in $n$. Bollob\\'{a}s confirmed the conjecture in 1977, and Erd\\H{o}s further asked for the exact extremal number. To the best of our knowledge, this problem has been solved only for all residues when $k\\leq 4$, and for $\\ell\\in \\{0,2\\}$ when $k\\geq 5$ is odd. In particular, Bai {\\it et al.} [arXiv:2503.03504] proved that if $G$ is an $n$-vertex graph with no cycle"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.02356","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/2606.02356/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"}