{"paper":{"title":"Boxicity and topological invariants","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Louis Esperet","submitted_at":"2015-03-19T13:50:31Z","abstract_excerpt":"The boxicity of a graph $G=(V,E)$ is the smallest integer $k$ for which there exist $k$ interval graphs $G_i=(V,E_i)$, $1 \\le i \\le k$, such that $E=E_1 \\cap \\cdots \\cap E_k$. In the first part of this note, we prove that every graph on $m$ edges has boxicity $O(\\sqrt{m \\log m})$, which is asymptotically best possible. We use this result to study the connection between the boxicity of graphs and their Colin de Verdi\\`ere invariant, which share many similarities. Known results concerning the two parameters suggest that for any graph $G$, the boxicity of $G$ is at most the Colin de Verdi\\`ere in"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1503.05765","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"}