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

We derive eigenvalue bounds for the distance-t 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 [New bounds on a hypercube coloring problem, Inform. Process. Lett. 84(5) (2002) 265–269], and improving their bound for several instances. We also apply the eigenvalue bounds to Lee graphs, extending results by Kim and Kim [The 2-distance coloring of the Cartesian product of cycles using optimal Lee codes, Discr. Appl. Math. 159(18) (2011) 2222–2228]. 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 spectral graph theory succeeds to capture the nature of the Lee metric.