The chromatic number of oriented graphs
Journal of Graph Theory
A coloring problem on the n-cube
Discrete Applied Mathematics
The Chromatic Number of Graph Powers
Combinatorics, Probability and Computing
Acyclic colorings of products of trees
Information Processing Letters
Acyclic coloring of graphs of maximum degree five: Nine colors are enough
Information Processing Letters
The 2-distance coloring of the Cartesian product of cycles using optimal Lee codes
Discrete Applied Mathematics
Distance coloring and distance edge-coloring of d- dimensional lattice
ICIC'12 Proceedings of the 8th international conference on Intelligent Computing Theories and Applications
A polyhedral study of the acyclic coloring problem
Discrete Applied Mathematics
Throughput analysis of multiple channel based wireless sensor networks
Wireless Networks
Note: A note on S-packing colorings of lattices
Discrete Applied Mathematics
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In this paper, we give a relatively simple though very efficient way to color the d-dimensional grid G(n1,n2,...,nd) (with ni vertices in each dimension 1 ≤ i ≤ d), for two different types of vertex colorings: (1) acyclic coloring of graphs, in which we color the vertices such that (i) no two neighbors are assigned the same color and (ii) for any two colors i and j, the subgraph induced by the vertices colored i or j is acyclic; and (2) k-distance coloring of graphs, in which every vertex must be colored in such a way that two vertices lying at distance less than or equal to k must be assigned different colors. The minimum number of colors needed to acyclically color (respectively k-distance color) a graph G is called acyclic chromatic number of G (respectively k-distance chromatic number), and denoted a(G) (respectively χk(G)).The method we propose for coloring the d-dimensional grid in those two variants relies on the representation of the vertices of Gd(n1,...,nd) thanks to its coordinates in each dimension; this gives us upper bounds on a(Gd(n1,...,nd)) and χk(Gd(n1,...,nd)).We also give lower bounds on a(Gd(n1,...,nd)) and χk(Gd(n1,...,nd)). In particular, we give a lower bound on a(G) for any graph G; surprisingly, as far as we know this result was never mentioned before. Applied to the d-dimensional grid Gd(n1,...,nd), the lower and upper bounds for a(Gd(n1,...,nd)) match (and thus give an optimal result) when the lengths in each dimension are "sufficiently large" (more precisely, if Σ i=1d 1/ni ≤ 1). If this is not the case, then these bounds differ by an additive constant at most equal to 1- ⌊Σi=1d1/ni⌋. Concerning χk(Gd(n1,...,nd)), we give exact results on its value for (1) k = 2 and any d ≥ 1, and (2) d = 2 and any k ≥ 1.In the case of acyclic coloring, we also apply our results to hypercubes of dimension d, Hd, which are a particular case of Gd(n1,...,nd) in which there are only 2 vertices in each dimension. In that case, the bounds we obtain differ by a multiplicative constant equal to 2.