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.

The general position problem on Kneser graphs and on some graph operations

“…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.…”

Characterization of general position sets and its applications to cographs and bipartite graphs

“…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.…”

Arcs in $$\mathbb Z^2_{2p}$$ Z 2 p 2

“…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.…”

“…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.…”

“…Four integers a, b, u, v with gcd(u, v)= 1 correspond to the line {(a +uk, b +vk) : k ∈ Z} on Z 2 . We define lines in Z 2 n to be images, by natural projection, of lines in the Z 2 (see Misiak et al 2016). We say that points a 1 , .…”