Distance domatic numbers for grid graphs

TL;DR

This paper determines the exact k-distance domatic numbers for grid graphs formed by the Cartesian product of two long paths, advancing understanding of multi-agent network arrangements.

Contribution

It provides the first exact values for the k-distance domatic number in grid graphs, a problem motivated by sensor and robotics network applications.

Findings

Exact k-distance domatic numbers for grid graphs are established.

Results apply to Cartesian products of two long paths.

Findings have implications for multi-agent network design.

Abstract

We say that a vertex-coloring of a graph is a proper k-distance domatic coloring if for each color, every vertex is within distance k from a vertex receiving that color. The maximum number of colors for which such a coloring exists is called the k-distance domatic number of the graph. The problem of determining the k-distance domatic number is motivated by questions about multi-agent networks including arrangements of sensors and robotics. Here, we find the exact k-distance domatic numbers for all grid graphs formed from the Cartesian product of two sufficiently long paths.