The rectilinear local crossing number of $K_n$

Abstract

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

arXiv commentChanges from v3: A few typos were corrected (ceiling symbols were missing)