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

AbstractLet Cm be the cycle of length m. We denote the Cartesian product of n copies of Cm by G(n,m):=Cm□Cm□⋯□Cm. The k-distance chromatic number χk(G) of a graph G is χ(Gk) where Gk is the kth power of the graph G=(V,E) in which two distinct vertices are adjacent in Gk 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 χ2(G(n,m)) for n=3 and m≥3. In particular, we compute exact values of χ2(G(3,m)) for 3≤m≤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 Z63 with minimum Lee distance 3 is 26