{"paper":{"title":"Improvements on the density of maximal 1-planar graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"G\\'eza T\\'oth, J\\'anos Bar\\'at","submitted_at":"2015-09-18T08:50:56Z","abstract_excerpt":"A graph is 1-planar if it can be drawn in the plane such that each edge is crossed at most once. A graph, together with a 1-planar drawing is called 1-plane. Brandenburg et al. showed that there are maximal 1-planar graphs with only $\\frac{45}{17}n + O(1)\\approx 2.647n$ edges and maximal 1-plane graphs with only $\\frac{7}{3}n+O(1)\\approx 2.33n$ edges. On the other hand, they showed that a maximal 1-planar graph has at least $\\frac{28}{13}n-O(1)\\approx 2.15n-O(1)$ edges, and a maximal 1-plane graph has at least $2.1n-O(1)$ edges.\n  We improve both lower bounds to $\\frac{20n}{9}\\approx 2.22n$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1509.05548","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"}