Plane Bichromatic Trees of Low Degree

TL;DR

The paper proves that for two disjoint point sets in the plane, there exists a non-crossing bipartite spanning tree with a degree bound that is optimal, solving a 1996 open problem.

Contribution

It establishes the existence of a low-degree non-crossing spanning tree in bipartite geometric graphs, providing the best possible degree bound.

Findings

Existence of a non-crossing spanning tree with bounded degree

Optimal upper bound on maximum degree proven

Solves an open problem from 1996

Abstract

Let $R$ and $B$ be two disjoint sets of points in the plane such that $|B|\leqslant |R|$, and no three points of $R\cup B$ are collinear. We show that the geometric complete bipartite graph $K(R,B)$ contains a non-crossing spanning tree whose maximum degree is at most $\max\left\{3, \left\lceil \frac{|R|-1}{|B|}\right\rceil + 1\right\}$; this is the best possible upper bound on the maximum degree. This solves an open problem posed by Abellanas et al. at the Graph Drawing Symposium, 1996.