Packing coloring of hypercubes with extended Hamming codes
Petr Gregor, Jaka Kranjc, Borut Lužar, Kenny Štorgel
TL;DR
This paper investigates the packing coloring of hypercubes by focusing on the packing chromatic number $\chi_\rho(Q_n)$. It introduces recursive, code-based constructions using extended Hamming codes $\hat{H}_m$ and their projections to build explicit packing colorings, yielding improved upper bounds for $\chi_\rho(Q_n)$ and establishing a corollary bound when $n=2^m$. The authors also address a question on packing colorings of Cartesian products, and discuss lower bounds and potential tightness through bipartition considerations, conjecturing $\chi_\rho(Q_n)=\chi_\rho^B(Q_n)$ for hypercubes. The work shows that leveraging structured codes within hypercubes can yield stronger upper bounds and informs future directions for exact values in higher dimensions and related graph products.
Abstract
A {\em packing coloring} of a graph $G$ is a mapping assigning a positive integer (a color) to every vertex of $G$ such that every two vertices of color $k$ are at distance at least $k+1$. The least number of colors needed for a packing coloring of $G$ is called the {\em packing chromatic number} of $G$. In this paper, we continue the study of the packing chromatic number of hypercubes and we improve the upper bounds reported by Torres and Valencia-Pabon ({\em P. Torres, M. Valencia-Pabon, The packing chromatic number of hypercubes, Discrete Appl. Math. 190--191 (2015), 127--140}) by presenting recursive constructions of subsets of distant vertices making use of the properties of the extended Hamming codes. We also answer in negative a question on packing coloring of Cartesian products raised by Brešar, Klavžar, and Rall ({\em Problem 5, Brešar et al., On the packing chromatic number of Cartesian products, hexagonal lattice, and trees. Discrete Appl. Math. 155 (2007), 2303--2311.}).