{"paper":{"title":"On the $4$-clique cover number of graphs","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Jialin He, Tianying Xie, Yihan Chen","submitted_at":"2025-06-12T08:30:34Z","abstract_excerpt":"In 1966, Erd\\H{o}s, Goodman, and P\\'osa proved that $\\lfloor n^2/4 \\rfloor$ cliques are sufficient to cover all edges in any $n$-vertex graph, with tightness achieved by the balanced complete bipartite graph. This result was generalized by Dau, Milenkovic, and Puleo, who showed that at most $\\lfloor \\frac n 3 \\rfloor \\lfloor \\frac {n+1} 3 \\rfloor \\lfloor \\frac {n+2} 3 \\rfloor$ cliques are needed to cover all triangles in any $n$-vertex graph $G$, and the bound is best possible as witnessed by the balanced complete tripartite graph. They further conjectured that for $t \\geq 4$, the $t$-clique c"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2506.10478","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/2506.10478/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"}