Distance Regular Colorings of $n$-Dimensional Rectangular Grid

TL;DR

This paper investigates the properties of distance regular colorings in n-dimensional rectangular grids, proving bounds on the number of colors and monotonicity of parameter matrices.

Contribution

It establishes that all irreducible distance regular colorings of n-dimensional grids have at most 2n+1 colors and characterizes the structure of their parameter matrices.

Findings

Parameter matrices form two monotonic sequences.

Maximum colors in irreducible colorings is 2n+1.

Distance regular colorings are linked to completely regular codes.

Abstract

We study the infinite graph of $n$-dimensional rectangular grid that doesn't appear distance regular and the distance regular colorings of this graph, which are defined as the distance colorings with respect to completely regular codes. It is proved that the elements of the parameter matrix of an arbitrary distance regular coloring form two monotonic sequences. It is shown that every irreducible distance regular coloring of the $n$-dimensional rectangular grid has at most $2n+1$ colors.