An Improvement of the Lower Bound on the Minimum Number of ≤<i>k</i>-Edges

• Main Authors: Javier Rodrigo, Susana Merchán, Danilo Magistrali, Mariló López
• Format: Article
• Language: English
• Published: MDPI AG 2021-03-01
• Series: Mathematics
• Subjects: combinatorial geometry; ≤k-edges; rectilinear crossing number; optimization; complete graphs
• Online Access: https://www.mdpi.com/2227-7390/9/5/525

In this paper, we improve the lower bound on the minimum number of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo> </mo><mo>≤</mo><mi>k</mi></mrow></semantics></math></inline-formula>-edges in sets of <i>n</i> points in general position in the plane when <i>k</i> is close to <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mfrac><mi>n</mi><mn>2</mn></mfrac></semantics></math></inline-formula>. As a consequence, we improve the current best lower bound of the rectilinear crossing number of the complete graph <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>K</mi><mi>n</mi></msub></semantics></math></inline-formula> for some values of <i>n</i>.