How many times can two minimum spanning trees cross?

2026-01-27

AI Summary

This study investigates the maximum number of crossings between minimum spanning trees (MSTs) induced by red and blue points in a bichromatic point set in the plane. For general point sets, the authors establish the first linear upper bound on the number of crossings between the two MSTs. Matching linear lower bounds are constructed for point sets in convex position and for dense configurations. Furthermore, under a uniformly random coloring model, the expected number of crossings is also shown to be linear. By integrating techniques from combinatorial geometry, probabilistic analysis, and graph theory, the work provides a rigorous modeling of crossing structures and derives tight asymptotic bounds, offering the first comprehensive characterization of the asymptotic behavior of crossing numbers in bichromatic MSTs.

Technology Category

• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Bridging structured and unstructured data
• Social Networks and Social Media: Computational social science

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.

Problem

• minimum spanning tree
• crossing number
• bicolored point set
• geometric graph
• combinatorial geometry

Innovation

• minimum spanning tree
• crossing number
• bicolored point set
• computational geometry
• random point set