The rectilinear local crossing number of $K_n$

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 ${\bar{\rm{lcr}}}(K_{14})=15$.Comment: 6 Figures. Changes from v1: Added keywords, MSC2010 codes, a single formula to consider all cases together, and the resolution of the case n=14 that remained as a conjecture on the previous version. Changes from v2: A minor error in Lemma 2 was corrected. Some typos were fixed. Figure 1 was eliminated and Figures 2 and 5 were improved slightly. The last section was split into two section