Distance Coloring and Distance Edge-Coloring of d- dimensional Lattice

Abstract

The d -dimensional lattice L _d is the Cartesian product of d two-way infinite paths. k -distance coloring (resp. k-distance edge-coloring) of G is a vertex coloring (resp. edge coloring) of G such that no two vertices (resp. edges) within distance k are given the same color, the minimum number of colors necessary to k -distance color (resp. k -distance edge-color) G, and is denoted by χ _k (G) (resp. ${\chi}^\prime_{k}(G)$). In this paper, we study the distance coloring and distance edge-coloring of d -dimensional lattice L _d , and give exact value of $\chi_{3}(L_{d}),~ {\chi}^\prime_{2}(L_{d})~{\rm and}~ {\chi}^\prime_{k}(L_{2})$ for any integers d ≥ 2, k ≥ 1.