Non-Monochromatic and Conflict-Free Coloring on Tree Spaces and Planar Network Spaces

TL;DR

This paper explores non-monochromatic and conflict-free coloring strategies for complex 1-dimensional spaces like tree spaces and planar network spaces, extending known interval coloring results.

Contribution

It introduces new coloring results for tree and planar network spaces, generalizing classical interval coloring theorems to more complex structures.

Findings

Non-monochromatic coloring with two colors in these spaces

Conflict-free coloring with three colors in these spaces

Extension of interval coloring results to complex 1D spaces

Abstract

It is well known that any set of n intervals in $\mathbb{R}^1$ admits a non-monochromatic coloring with two colors and a conflict-free coloring with three colors. We investigate generalizations of this result to colorings of objects in more complex 1-dimensional spaces, namely so-called tree spaces and planar network spaces.