Acyclic and k-distance coloring of the grid
Résumé
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 \leq i \leq 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 k(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 k(Gd (n1, . . . , nd )). We also give lower bounds on a(Gd (n1, . . . , nd )) and k(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 sum_{i=1}^{d} (1/ni)\leq 1). If this is not the case, then these bounds differ by an additive constant at most equal to 1−⌊sum_{i=1}^{d} (1/ni)⌋. Concerning k(Gd (n1, . . . , nd )), we give exact results on its value for (1) k = 2 and any d \geq 1, and (2) d = 2 and any k \geq 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.
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