On the geometric $k$-colored crossing number of $K_n$

TL;DR

This paper investigates the minimal monochromatic crossings in colored drawings of complete graphs, improving bounds for small k by developing a method to generate low-crossing drawings for larger graphs using heuristic search.

Contribution

It introduces a new procedure for constructing $k$-edge colored drawings of $K_n$ with fewer crossings, improving asymptotic bounds for $k=2$ to $10$.

Findings

Improved asymptotic upper bounds for $ar{ar{ ext{cr}}}_k(K_n)$ for $k=2$ to $10$.

Development of a general method to generate low-crossing drawings for larger $n$.

Use of heuristic search and MAX-$k$-CUT formulation to optimize drawings.

Abstract

We study the \emph{geometric $k$-colored crossing number} of complete graphs $\overline{\overline{\text{cr}}}_k(K_n)$, which is the smallest number of monochromatic crossings in any $k$-edge colored straight-line drawing of $K_n$. We substantially improve asymptotic upper bounds on $\overline{\overline{\text{cr}}}_k(K_n)$ for $k=2,\ldots, 10$ by developing a procedure for general $k$ that derives $k$-edge colored drawings of $K_n$ for arbitrarily large $n$ from initial drawings with a low number of monochromatic crossings. We obtain the latter by heuristic search, employing a \textsc{MAX-$k$-CUT}-formulation of a subproblem in the process.