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 ${\bar{\rm{lcr}}}(K_{14})=15$.

Figures (5)

• Figure 1: The absolute difference of points between the shaded and not shaded regions is at most $(n-2)/3$ (left) or $\textcolor{black}{\lceil(n-3)/3\rceil}$ (right).
• Figure 2: (a) $\triangle q_1q_2q_3 \cup S(q_1r_0q_2)$ has more than $n-3$ points of $P \setminus \{q_1,q_2,q_3 \}$. (bc) $S(q_1pq_2)$ is a proper subset of $S(q_1r_0q_2)$.
• Figure 3: The number of crossings of edges in the same $C_i$ or between $C_i$ and $C_{i+1}$
• Figure 4: The construction when $n=3k+8$. Each of the five thick dashed lines joins the endpoints of the corresponding arc.
• Figure 5: The construction for $n\in \{ 11, 14, 17 \}$. The 11-point set consists of the black points, the 14-point set consists of the black or white points, and the 17-point set consists of all of the points shown.

Theorems & Definitions (9)

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