Spanning Trees of Multicoloured Point Sets with Few Intersections

Abstract

Kano et al. proved that if P ₀, P ₁, ..., P _{k − − 1} are pairwise disjoint collections of points in general position, then there exist spanning trees T ₀, T ₁, ..., T _{k − − 1}, of P ₀, P ₁, ..., P _{k − − 1}, respectively, such that the edges of T ₀ ∪ T ₁ ∪ ... ∪ T _k − 1 intersect in at most (k – 1)n – k(k – 1)/2 points. In this paper we show that this result is asymptotically tight within a factor of 3/2. To prove this, we consider alternating collections, that is, collections such that the points in P: = P ₀ ∪ P ₁ ∪ ... ∪ P _k − 1 are in convex position, and the points of the P _i ’s alternate in the convex hull of P.