2022/03/11 by Oswin Aichholzer, Aichholzer, Oswin, Alfredo García +7 · 3 citations
Computer Science · Engineering · #Computational Geometry (cs.CG) #Computational Geometry and Mesh Generation #FOS: Computer and information sciences #VLSI and FPGA Design Techniques #graph theory and CDMA systems
paper · pdf · doi:10.48550/arxiv.2203.06143
openalex publication_date 2022/03/11 · openalex created_date 2022/04/03 · openalex updated_date 2026/07/28
Simple drawings are drawings of graphs in which the edges are Jordan arcs and each pair of edges share at most one point (a proper crossing or a common endpoint). We introduce a special kind of simple drawings that we call generalized twisted drawings. A simple drawing is generalized twisted if there is a point O such that every ray emanating from O crosses every edge of the drawing at most once and there is a ray emanating from O which crosses every edge exactly once. Via this new class of simple drawings, we show that every simple drawing of the complete graph with n vertices contains Ω(n(1)/(2)) pairwise disjoint edges and a plane path of length Ω((log n )/(log log n)). Both results improve over previously known best lower bounds. On the way we show several structural results about and properties of generalized twisted drawings. We further present different characterizations of generalized twisted drawings, which might be of independent interest.