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

초록

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(m)square C(m)square ... square C(m). The k-distance chromatic number chi(k)(G) of a graph G is chi (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 chi(2)(G(n. m)) for n = 3 and m >= 3. In particular, we compute exact values of chi(2)(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 Z(6)(3) with minimum Lee distance 3 is 26. (C) 2011 Elsevier B.V. All rights reserved.

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