The 2-distance coloring of the Cartesian product of cycles using optimal Lee codes

Let C"m be the cycle of length m. We denote the Cartesian product of n copies of C"m by G(n,m):=C"[email protected]?C"[email protected][email protected]?C"m. The k-distance chromatic number @g"k(G) of a graph G is @g(G^k) where G^k is the kth power of the graph G=(V,E) in which two distinct vertices are adjacent in G^k if and only if their distance in G is at most k. The k-distance chromatic number of G(n,m) is related to optimal codes over the ring of integers modulo m with minimum Lee distance k+1. In this paper, we consider @g"2(G(n,m)) for n=3 and m>=3. In particular, we compute exact values of @g"2(G(3,m)) for [email protected][email protected]?8 and m=4k, and upper bounds for m=3k or m=5k, for any positive integer k. We also show that the maximal size of a code in Z"6^3 with minimum Lee distance 3 is 26.