{"paper":{"title":"Four-Cycle Counting in Low-Degeneracy Graph Streams","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"cs.DS","authors_text":"Pan Peng, Sebastian L\\\"uderssen, Stefan Neumann","submitted_at":"2026-06-16T09:49:45Z","abstract_excerpt":"We study the problem of $(1+\\varepsilon)$-approximating the number of four-cycles in graphs given as arbitrary order edge streams. We propose two new algorithms based on sampling induced subgraphs. Our first contribution is a two-pass algorithm that uses $\\widetilde{O}(\\kappa m / \\sqrt{T})$ space, where $m$ is the number of edges, $T$ is the number of four-cycles, and $\\kappa$ is the graph's degeneracy. This algorithm improves upon existing theoretical bounds and is provably optimal for constant-degeneracy graphs, matching the known $\\Omega(m/\\sqrt{T})$ lower bound up to lower-order factors. O"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.17732","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/2606.17732/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"}