Eigenvalue bounds for the distance-$t$ chromatic number of a graph and their application to Lee codes

TL;DR

This paper develops eigenvalue bounds for the t-distance chromatic number of graphs, applies them to hypercube and Lee graphs, and characterizes perfect Lee codes of minimum distance 3, introducing spectral methods to Lee codes.

Contribution

It introduces the first spectral bounds for Lee codes and extends eigenvalue techniques to analyze the t-distance chromatic number in graphs.

Findings

Eigenvalue bounds improve bounds for hypercube graphs.

Spectral methods successfully characterize perfect Lee codes.

New bounds extend previous results by Kim and Kim.

Abstract

We derive eigenvalue bounds for the $t$-distance chromatic number of a graph, which is a generalization of the classical chromatic number. We apply such bounds to hypercube graphs, providing alternative spectral proofs for results by Ngo, Du and Graham [Inf. Process. Lett., 2002], and improving their bound for several instances. We also apply the eigenvalue bounds to Lee graphs, extending results by Kim and Kim [Discrete Appl. Math., 2011]. Finally, we provide a complete characterization for the existence of perfect Lee codes of minimum distance $3$. In order to prove our results, we use a mix of spectral and number theory tools. Our results, which provide the first application of spectral methods to Lee codes, illustrate that such methods succeed to capture the nature of the Lee metric.