# On the 2-colored crossing number

## TL;DR

This paper investigates the rectilinear 2-colored crossing number of complete graphs, establishing bounds and ratios, and introduces methods to find optimal instances using heuristics and integer programming.

## Contribution

It provides new bounds for the rectilinear 2-colored crossing number of complete graphs and improves ratio bounds for fixed drawings, combining theoretical and computational approaches.

## Key findings

- Established lower and upper bounds for $K_n$
- Derived asymptotic bounds from small instances
- Improved ratio bounds for fixed drawings of $K_n$

## Abstract

Let $D$ be a straight-line drawing of a graph. The rectilinear 2-colored crossing number of $D$ is the minimum number of crossings between edges of the same color, taken over all possible 2-colorings of the edges of $D$. First, we show lower and upper bounds on the rectilinear 2-colored crossing number for the complete graph $K_n$. To obtain this result, we prove that asymptotic bounds can be derived from optimal and near-optimal instances with few vertices. We obtain such instances using a combination of heuristics and integer programming. Second, for any fixed drawing of $K_n$, we improve the bound on the ratio between its rectilinear 2-colored crossing number and its rectilinear crossing number.