The k-Distance Independence Number and 2-Distance Chromatic Number of Cartesian Products of Cycles

Abstract

A set $S \subseteq V(G)$ is a k-distance independent set of a graph G if the distance between every two vertices of S is greater than k. The k-distance independence number $\alpha _k(G)$ of G is the maximum cardinality over all k-distance independent sets in G. A k-distance coloring of G is a function f from V(G) onto a set of positive integers (colors) such that for any two distinct vertices u and v at distance less than or equal to k we have $f(u) \not = f(v)$. The k-distance chromatic number of a graph G is the smallest number of colors needed to have a k-distance coloring of G. The k-distance independence numbers and 2-distance chromatic numbers of Cartesian products of cycles are considered. A computer-aided method with the isomorphic rejection is used to determine the k-distance independence numbers of Cartesian products of cycles. By using these results, several lower and upper bounds on the maximal cardinality $A^L_q(n, d)$ of a q-ary Lee code of length n with a minimum distance at least d are improved. It is also established that the 2-distance chromatic number of G equals $\left\lceil \frac{|V(G)|}{\alpha (G^2)} \right\rceil$ for $G=C_m \Box C_n \Box C_k$, whenever $k \ge 3$ and $(m,n)\in \{(3,3), (3,4), (3,5), (4,4)\}$ or k, m and n are all multiples of seven. Moreover, it is shown that the 2-distance chromatic number of the three-dimensional square lattice is equal to seven.