Separation Dimension and Chromatic Bounds for Fractal Partitions
The Separation Dimension of a partition of an Ahlfors-regular fractal is the Hausdorff dimension of its Separation Set, the union of all interfaces between tiles. For Geometrically Regular Partitions, the Chromatic Tiling Theorem bounds the chromatic number of the tiles' adjacency graph by a power law in the tile-size ratio whose exponent is the Separation Dimension, proved by a Hausdorff-measure ball-packing argument and Brooks' theorem. This shows the coloring complexity of a fractal tiling is governed by the dimension of its interfaces, not of the fractal itself.
THE CHROMATIC TILING THEOREM: SCALING LAWS AND THE SEPARATION DIMENSION OF FRACTAL PARTITIONS ROBIN
Introduces the Separation Dimension of a partition of an Ahlfors d-regular fractal: the Hausdorff dimension of the Separation Set, the union of all interfaces between the tiles. Restricting to Geomet…