Abstract. In this paper we show that at most 2 gcd(m, n) points can be placed with no three in a line on an m × n discrete torus. In the situation when gcd(m, n) is a prime, we completely solve the problem.
| and we call it the weight of f in G. The minimum weight of a k R DF of G is called the k-rainbow domination number of G and it is denoted by γ rk (G). We investigate the 2-rainbow domination number of Cartesian products of cycles. We give the exact value of the 2-rainbow domination number of C n C 3 and we give the estimation of this number with respect to C n C 5 , (n ≥ 3). Additionally, for n = 3, 4, 5, 6, we show that γ r 2 (C n C 5 ) = 2n.
We prove that any compact simply connected manifold carrying a structure of Riemannian 3-or 4-symmetric space is formal in the sense of Sullivan. This result generalizes Sullivan's classical theorem on the formality of symmetric spaces, but the proof is of a different nature, since for generalized symmetric spaces techniques based on the Hodge theory do not work. We use the Thomas theory of minimal models of fibrations and the classification of 3-and 4-symmetric spaces.
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