On covering and coloring problems for rook domains

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Given the set Vkn of all vectors with length n and components 0, 1,...,k-1 from the ring of the integers modulo k, the Hamming distance H(X, Y) between X, Y ε{lunate} Vkn is defined as the number of components in which X and Y differ, and the j-dimensional rook domain of X ε{lunate} Vkn is defined as the set of vectors in Vkn within distance j and X. A subset H of Vkn is a (n, k, s)-covering set if Vkn can be obtained as the union of the (n-s)-dimensional rook domains of the vectors in H. The covering problem for Vkn consists of the determination of the minimal cardinality γ(n, k, s) of such a subset. The search for the maximal number of disjoint (n, k, s)-covering sets is known as the coloring problem for Vkn and this number is denoted by δ(n, k, s). In this paper we obtain some general bounds for the functions γ(n, k, s) and δ(n, k, s), and calculate some of their values for the cases s ≤ n - 2. © 1985.




Carnielli, W. A. (1985). On covering and coloring problems for rook domains. Discrete Mathematics, 57(1–2), 9–16. https://doi.org/10.1016/0012-365X(85)90152-9

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