{"paper":{"title":"Strong cliques and forbidden cycles","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Fran\\c{c}ois Pirot, Ross J. Kang, Wouter Cames van Batenburg","submitted_at":"2019-03-14T15:45:20Z","abstract_excerpt":"Given a graph $G$, the strong clique number $\\omega_2'(G)$ of $G$ is the cardinality of a largest collection of edges every pair of which are incident or connected by an edge in $G$. We study the strong clique number of graphs missing some set of cycle lengths. For a graph $G$ of large enough maximum degree $\\Delta$, we show among other results the following: $\\omega_2'(G)\\le5\\Delta^2/4$ if $G$ is triangle-free; $\\omega_2'(G)\\le3(\\Delta-1)$ if $G$ is $C_4$-free; $\\omega_2'(G)\\le\\Delta^2$ if $G$ is $C_{2k+1}$-free for some $k\\ge 2$. These bounds are attained by natural extremal examples. Our wo"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1903.06087","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":""},"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"}