Polychromatic Colorings on the Hypercube

TL;DR

This paper investigates the maximum number of colors for edge colorings of hypercubes ensuring every embedding of a subgraph contains all colors, introducing new methods and applying them to punctured hypercubes and larger subcubes.

Contribution

It introduces structured colorings called simple colorings and develops techniques to determine polychromatic numbers for complex hypercube subgraphs.

Findings

Determined polychromatic numbers for punctured hypercubes.

Developed a translation of coloring problems into grid shape sequences.

Presented new constructions for coloring larger subcubes.

Abstract

Given a subgraph G of the hypercube Q_n, a coloring of the edges of Q_n such that every embedding of G contains an edge of every color is called a G-polychromatic coloring. The maximum number of colors with which it is possible to G-polychromatically color the edges of any hypercube is called the polychromatic number of G. To determine polychromatic numbers, it is only necessary to consider a structured class of colorings, which we call simple. The main tool for finding upper bounds on polychromatic numbers is to translate the question of polychromatically coloring the hypercube so every embedding of a graph G contains every color into a question of coloring the 2-dimensional grid so that every so-called shape sequence corresponding to G contains every color. After surveying the tools for finding polychromatic numbers, we apply these techniques to find polychromatic numbers of a class of graphs called punctured hypercubes. We also consider the problem of finding polychromatic numbers in the setting where larger subcubes of the hypercube are colored. We exhibit two new constructions which show that this problem is not a straightforward generalization of the edge coloring problem.