2016
DOI: 10.1016/j.disc.2015.08.006
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A note on the no-three-in-line problem on a torus

Abstract: 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.

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Cited by 21 publications
(13 citation statements)
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“…This concept was introduced-under the present name-in [12] in part motivated by the Dudeney's 1917 no-three-in-line problem [5] (see [10,14,17] for recent related results) and by a corresponding problem in discrete geometry known as the general position subset selection problem [7,16]. Independently geodetic irredundant sets were earlier introduced in [18], a concept which is equivalent to the general position sets.…”
Section: Introductionmentioning
confidence: 99%
“…This concept was introduced-under the present name-in [12] in part motivated by the Dudeney's 1917 no-three-in-line problem [5] (see [10,14,17] for recent related results) and by a corresponding problem in discrete geometry known as the general position subset selection problem [7,16]. Independently geodetic irredundant sets were earlier introduced in [18], a concept which is equivalent to the general position sets.…”
Section: Introductionmentioning
confidence: 99%
“…Motivated by the century old Dudeney's no-three-in-line problem [6] (see [11,14,17] for recent developments on it) and by the general position subset selection problem [7,16] from discrete geometry, the natural related problem was introduced to graph theory in [12] as follows. Let G = (V (G), E(G)) be a graph.…”
Section: Introductionmentioning
confidence: 99%
“…It is easy to see that m(2, q) is an upper bound for τ (Z 2 p ) in the case of a prime q = p. To establish the lower bound for τ (Z 2 p ), recall that there are arcs of Z 2 p of cardinality p + 1 (see for instance Misiak et al 2016 for more details). Therefore, the following lemma is true.…”
Section: Basic Factsmentioning
confidence: 99%
“…This modified problem is still interesting and was investigated in Huizenga (2006), Kurz (2009) and Misiak et al (2016). Many authors considered arcs in the context of projective geometry, see e.g.…”
Section: Introductionmentioning
confidence: 99%
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