Empty Triangles in Generalized Twisted Drawings of $$K_n$$

García, Alfredo; Tejel, Javier; Vogtenhuber, Birgit; Weinberger, Alexandra

doi:10.1007/978-3-031-22203-0_4

Lecture Notes in Computer Science

2023

DOI: 10.1007/978-3-031-22203-0_4

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Empty Triangles in Generalized Twisted Drawings of $$K_n$$

Alfredo García

Javier Tejel

Birgit Vogtenhuber

et al.

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2024

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Twisted Ways to Find Plane Structures in Simple Drawings of Complete Graphs

Aichholzer,

García,

Tejel

et al. 2024

Discrete Comput Geom

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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). A simple drawing is c-monotone if there is a point O such that each ray emanating from O crosses each edge of the drawing at most once. We introduce a special kind of c-monotone drawings that we call generalized twisted drawings. A c-monotone drawing is generalized twisted if there is a ray emanating from O that crosses all the edges of the drawing. Via this class of drawings, we show that every simple drawing of the complete graph with n vertices contains $$\Omega (n^{\frac{1}{2}})$$ Ω ( n 1 2 ) pairwise disjoint edges and a plane cycle (and hence path) of length $$\Omega (\frac{\log n }{\log \log n})$$ Ω ( log n log log n ) . Both results improve over best previously published lower bounds. On the way we show several structural results and properties of generalized twisted and c-monotone drawings, some of which we believe to be of independent interest. For example, we show that a drawing D is c-monotone if there exists a point O such that no edge of D is crossed more than once by any ray that emanates from O and passes through a vertex of D.

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Twisted Ways to Find Plane Structures in Simple Drawings of Complete Graphs

Aichholzer,

García,

Tejel

et al. 2024

Discrete Comput Geom

Self Cite

View full text Add to dashboard Cite

show abstract

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Empty Triangles in Generalized Twisted Drawings of $$K_n$$

Cited by 1 publication

References 12 publications

Twisted Ways to Find Plane Structures in Simple Drawings of Complete Graphs

Twisted Ways to Find Plane Structures in Simple Drawings of Complete Graphs

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