A set D of vertices in a graph is said t o be a dominating set if every vertex not in D is adjacent to some vertex in D. The domination number P(G) of a graph G is the size of a smallest dominating set. G is called domination balanced if its vertex set can be partitioned into P(G) subsets SO that each subset is a smallest dominating set of the complement G of G. The purpose of this paper is to characterize these graphs.
We prove that there does not exist a tiling with Lee spheres of radius at least 2 in the 3-dimensional Euclidean space. In particular, this result verifies a conjecture of Golomb and Welch for n = 3.
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