Linear $d$-polychromatic $Q_{d-1}$-colorings of the Hypercube

TL;DR

This paper investigates linear colorings of hypercube substructures, establishing exact bounds for the maximum number of colors in certain polychromatic colorings and providing computational results for specific cases.

Contribution

It proves that for all d ≥ 3, the maximum number of colors in a linear d-polychromatic (d-1)-subcube coloring is 2, and computes bounds for other parameters using computer search.

Findings

For all d ≥ 3, p_{lin}^{d-1}(d) = 2.

Determined p_{lin}^ ext{ell}(d) for specific values of ell and d.

Improved lower bounds for certain coloring parameters.

Abstract

Let $n \ge d \ge \ell \ge 1$ be integers, and denote the $n$-dimensional hypercube by $Q_n$. A coloring of the $\ell$-dimensional subcubes $Q_\ell$ in $Q_n$ is called a $Q_\ell$-coloring. Such a coloring is $d$-polychromatic if every $Q_d$ in the $Q_n$ contains a $Q_\ell$ of every color. In this paper we consider a specific class of $Q_\ell$-colorings that are called linear. Given $\ell$ and $d$, let $p_{lin}^\ell(d)$ be the largest number of colors such that there is a $d$-polychromatic linear $Q_\ell$-coloring of $Q_n$ for all $n \ge d$. We prove that for all $d \ge 3$, $p_{lin}^{d-1}(d) = 2$. In addition, using a computer search, we determine $p_{lin}^\ell(d)$ for some specific values of $\ell$ and $d$, in some cases improving on previously known lower bounds.