3--symmetric and 3--decomposable drawings of $K_n$ (extended version)

TL;DR

This paper investigates the structure of crossing-minimal geometric drawings of complete graphs, focusing on 3-symmetry and 3-decomposability, and provides new bounds and constructions that improve existing crossing number estimates.

Contribution

It proves lower bounds for 3-decomposable drawings and constructs new 3-symmetric and 3-decomposable drawings that improve the upper bounds of the rectilinear crossing number of $K_n$.

Findings

Any 3-decomposable drawing has at least 0.380029 * binomial(n,4) crossings.

Constructed 3-symmetric and 3-decomposable drawings improve the upper bound to 0.380488 * binomial(n,4).

Explicit constructions for n<100 match or surpass previous best known drawings.

Abstract

Even the most superficial glance at the vast majority of crossing-minimal geometric drawings of $K_n$ reveals two hard-to-miss features. First, all such drawings appear to be 3-fold symmetric (or simply {\em 3-symmetric}) . And second, they all are {\em 3-decomposable}, that is, there is a triangle $T$ enclosing the drawing, and a balanced partition $A, B, C$ of the underlying set of points $P$, such that the orthogonal projections of $P$ onto the sides of $T$ show $A$ between $B$ and $C$ on one side, $B$ between $A$ and $C$ on another side, and $C$ between $A$ and $B$ on the third side. In fact, we conjecture that all optimal drawings are 3-decomposable, and that there are 3-symmetric optimal constructions for all $n$ multiple of 3. In this paper, we show that any 3-decomposable geometric drawing of $K_n$ has at least $0.380029\binom{n}{4}+\Theta(n^3)$ crossings. On the other hand, we produce 3-symmetric and 3-decomposable drawings that improve the {\em general} upper bound for the rectilinear crossing number of $K_n$ to $0.380488\binom{n}{4}+\Theta(n^3)$. We also give explicit 3-symmetric and 3-decomposable constructions for $n<100$ that are at least as good as those previously known.