On Robust Colorings of Hamming-Distance Graphs

TL;DR

This paper investigates the chromatic number of Hamming-distance graphs and introduces a notion of robustness for 4-colorings of specific graphs, providing explicit descriptions of maximally robust colorings.

Contribution

It presents the chromatic numbers for various parameters and introduces a new robustness concept, with explicit descriptions for maximally robust 4-colorings of certain graphs.

Findings

Chromatic numbers for various $(q,n,d)$ parameters.

A new robustness measure for colorings based on color swapping tolerance.

Explicit characterization of maximally robust 4-colorings for $H_2(n,n-1)$.

Abstract

$H_q(n,d)$ is defined as the graph with vertex set ${\mathbb Z}_q^n$ and where two vertices are adjacent if their Hamming distance is at least $d$. The chromatic number of these graphs is presented for various sets of parameters $(q,n,d)$. For the $4$-colorings of the graphs $H_2(n,n-1)$ a notion of robustness is introduced. It is based on the tolerance of swapping colors along an edge without destroying properness of the coloring. An explicit description of the maximally robust $4$-colorings of $H_2(n,n-1)$ is presented.