How many times can two minimum spanning trees cross?

TL;DR

This paper investigates the maximum number of crossings between two minimum spanning trees in a bipartite point set, establishing linear bounds in various configurations and analyzing the expected crossings in random scenarios.

Contribution

It introduces the bicolored MST crossing number, provides linear bounds for generic, dense, and convex point sets, and analyzes the expected crossings in random point and coloring distributions.

Findings

Linear upper bound for generic point sets

Linear lower bounds for dense and convex point sets

Expected crossings are linear for random points and colorings

Abstract

Let $P$ be a generic set of $n$ points in the plane, and let $P=R\cup B$ be a coloring of $P$ in two colors. We are interested in the number of crossings between the minimum spanning trees (MSTs) of $R$ and $B$, denoted by $\crossAB(R,B)$. We define the \emph{bicolored MST crossing number} of $P$, denoted by $\cross(P)$, as $\cross(P) = \max_{P= R\cup B}(\crossAB(R,B))$. We prove a linear upper bound for $\cross(P)$ when $P$ is generic. If $P$ is dense or in convex position, we provide linear lower bounds. Lastly, if $P$ is chosen uniformly at random from the unit square and is colored uniformly at random, we prove that the expected value of $\crossAB(R,B)$ is linear.