{"paper":{"title":"On the Density of non-Simple 3-Planar Graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DS"],"primary_cat":"cs.CG","authors_text":"Chrysanthi N. Raftopoulou, Michael A. Bekos, Michael Kaufmann","submitted_at":"2016-02-16T11:45:25Z","abstract_excerpt":"A \\emph{$k$-planar graph} is a graph that can be drawn in the plane such that every edge is crossed at most $k$ times. For $k \\leq 4$, Pach and T\\'oth proved a bound of $(k+3)(n-2)$ on the total number of edges of a $k$-planar graph, which is tight for $k=1,2$. For $k=3$, the bound of $6n-12$ has been improved to $\\frac{11}{2}n-11$ and has been shown to be optimal up to an additive constant for simple graphs. In this paper, we prove that the bound of $\\frac{11}{2}n-11$ edges also holds for non-simple $3$-planar graphs that admit drawings in which non-homotopic parallel edges and self-loops are"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1602.04995","kind":"arxiv","version":3},"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"}