{"paper":{"title":"Set Representations of Linegraphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Jun-Lin Guo, Tao-Ming Wang, Ton Kloks, Yue-Li Wang","submitted_at":"2013-09-01T00:50:58Z","abstract_excerpt":"Let $G$ be a graph with vertex set $V(G)$ and edge set $E(G)$. A family $\\mathcal{S}$ of nonempty sets $\\{S_1,\\ldots,S_n\\}$ is a set representation of $G$ if there exists a one-to-one correspondence between the vertices $v_1, \\ldots, v_n$ in $V(G)$ and the sets in $\\mathcal{S}$ such that $v_iv_j \\in E(G)$ if and only if $S_i\\cap S_j\\neq \\es$. A set representation $\\mathcal{S}$ is a distinct (respectively, antichain, uniform and simple) set representation if any two sets $S_i$ and $S_j$ in $\\mathcal{S}$ have the property $S_i\\neq S_j$ (respectively, $S_i\\nsubseteq S_j$, $|S_i|=|S_j|$ and $|S_i\\"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1309.0170","kind":"arxiv","version":2},"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"}