The rectilinear local crossing number of $K_n$

TL;DR

This paper precisely determines the rectilinear local crossing number of complete graphs for all n, providing explicit formulas and exact values for specific cases, advancing understanding of geometric graph crossing properties.

Contribution

The paper derives exact formulas for the rectilinear local crossing number of K_n for all n, including special cases, filling a gap in geometric graph theory.

Findings

Explicit formula for n not in {8,14}

Exact values for K_8 and K_14

Complete characterization of rectilinear local crossing numbers

Abstract

We determine ${\bar{\rm{lcr}}}(K_n)$, the rectilinear local crossing number of the complete graph $K_n$ for every $n$. More precisely, for every $n \notin \{8, 14 \}, $ \[ {\bar{\rm{lcr}}}(K_n)=\left\lceil \frac{1}{2} \left( n-3-\left\lceil \frac{n-3}{3} \right\rceil \right) \left\lceil \frac{n-3}{3} \right\rceil \right\rceil, \] ${\bar{\rm{lcr}}}(K_8)=4$, and ${\bar{\rm{lcr}}}(K_{14})=15$.