# Non-crossing geometric spanning trees with bounded degree and monochromatic leaves on bicolored point sets

## TL;DR

This paper proves the existence of non-crossing geometric spanning trees on bicolored point sets with degree bounds on red points and all blue points as leaves, under certain size constraints.

## Contribution

It introduces a method to construct non-crossing spanning trees with degree bounds and monochromatic leaves on bicolored point sets, extending previous geometric graph results.

## Key findings

- Existence of such trees under specified conditions
- Construction method for non-crossing spanning trees with degree constraints
- Application to geometric graph theory problems

## Abstract

Let $R$ and $B$ be a set of red points and a set of blue points in the plane, respectively, such that $R\cup B$ is in general position, and let $f:R \to \{2,3,4, \ldots \}$ be a function. We show that if $2\le |B|\le \sum_{x\in R}(f(x)-2) + 2$, then there exists a non-crossing geometric spanning tree $T$ on $R\cup B$ such that $2\le \operatorname{deg}_T(x)\le f(x)$ for every $x\in R$ and the set of leaves of $T$ is $B$, where every edge of $T$ is a straight-line segment.