The chromatic number of the square of the 8-cube

TL;DR

This paper determines the chromatic number of the square of the 8-cube as 13, using coding theory concepts, and classifies the symmetries of such colorings, advancing understanding of cube-like graph colorings.

Contribution

It solves the open problem of the chromatic number of Q_8^2 by establishing it as 13 and classifies symmetric colorings, linking graph coloring with coding theory.

Findings

The chromatic number of Q_8^2 is 13.

Existence of specific symmetric 13-colorings.

Classification of these colorings based on symmetry.

Abstract

A cube-like graph is a Cayley graph for the elementary abelian group of order $2^n$. In studies of the chromatic number of cube-like graphs, the $k$th power of the $n$-dimensional hypercube, $Q_n^k$, is frequently considered. This coloring problem can be considered in the framework of coding theory, as the graph $Q_n^k$ can be constructed with one vertex for each binary word of length $n$ and edges between vertices exactly when the Hamming distance between the corresponding words is at most $k$. Consequently, a proper coloring of $Q_n^k$ corresponds to a partition of the $n$-dimensional binary Hamming space into codes with minimum distance at least $k+1$. The smallest open case, the chromatic number of $Q_8^2$, is here settled by finding a 13-coloring. Such 13-colorings with specific symmetries are further classified.