Algorithms for Coloring Quadtrees

TL;DR

This paper presents efficient algorithms for coloring quadtrees with minimal colors, ensuring no adjacent squares share the same color, with tight bounds established for different adjacency definitions and quadtree types.

Contribution

It introduces simple linear time algorithms for coloring quadtrees under various adjacency definitions, establishing tight bounds for the number of colors needed.

Findings

Balanced quadtrees can be colored with three colors when sharing sides.

Unbalanced quadtrees require four colors for side adjacency.

Six colors suffice when adjacency includes corners, with some requiring at least five.

Abstract

We describe simple linear time algorithms for coloring the squares of balanced and unbalanced quadtrees so that no two adjacent squares are given the same color. If squares sharing sides are defined as adjacent, we color balanced quadtrees with three colors, and unbalanced quadtrees with four colors; these results are both tight, as some quadtrees require this many colors. If squares sharing corners are defined as adjacent, we color balanced or unbalanced quadtrees with six colors; for some quadtrees, at least five colors are required.