How many times can two minimum spanning trees cross?

TL;DR

The paper introduces the bicolored MST crossing number, cr-MST(P), as a geometric proxy for spatial mixing between two colored point sets and establishes a linear upper bound for generic point sets. It then derives linear lower bounds in several structured cases (convex position, grid perturbations) and shows that random point sets with random colorings have an expected linear number of crossings. Additionally, the work analyzes dense point sets and proves a linear lower bound dependent on density, linking geometric distribution to cross-structure complexity. Overall, the results significantly advance understanding of MST interactions under two-colorings and suggest directions for tight bounds, extensions to non-generic configurations, and computational complexity questions.

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.

Figures (16)

• Figure 1: A set of $5$ points with bicolored MST crossing number equal to $0$. The middle and right pictures show the two MSTs for two different colorings. Interestingly, removing the top point increases the bicolored MST crossing number to $1$.
• Figure 2: Proof of Lemma \ref{['lem-crossingssmallangle']}.
• Figure 3: Colorings of the proof of Theorem \ref{['thm-genlowertwocols']}.
• Figure 4: Proof of Lemma \ref{['lem-islandsdonotinteract']}. The wedge is drawn with a larger angle for readability.
• Figure 5: Situation after the $i^{th}$ step of the procedure in the proof of Theorem \ref{['thm-genlowerthreecols']}. Wedge $W_i$ is drawn with a larger angle for readability.

Theorems & Definitions (61)

• theorem 1
• theorem 2
• theorem 3
• theorem 4
• theorem 5
• theorem 6
• theorem 7
• lemma 1
• proof
• lemma 2