Progress in the no-three-in-line-problem

Flammenkamp, Achim

doi:10.1016/0097-3165(92)90012-j

Cited by 17 publications

(8 citation statements)

References 8 publications

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“…The classical Dudeney's no-three-in-line problem [3,4,7] is to determine the largest vertex number that can be placed in the m × m grid such that no three vertices lie on a line. The problem has been further studied in several recent papers [10,12,15,17].…”

Section: Introductionmentioning

confidence: 99%

On the general position set of two classes of graphs

Yao,

He,

et al. 2020

Preprint

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The general position problem is to find the cardinality of a largest vertex subset S such that no triple of vertices of S lie on a common geodesic. For a connected graph G, the cardinality of S is denoted by gp(G) and called gp-number (or general position number) of G. In the paper, we obtain an upper bound and a lower bound regarding gp-number in all cactus with k cycles and t pendant edges. Furthermore, the gp-number of wheel graph is determined.

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Section: Introductionmentioning

confidence: 99%

On the general position set of two classes of graphs

Yao,

He,

et al. 2020

Preprint

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“…The problem of finding for which n this value is reached is known as the no-three-in-line problem. For a short concise overview of the historical development, see [1], [2].Usually the solutions are partitioned into the following eight symmetry classes: These having full symmetry (abbreviation, full), having only rotational symmetry (half rotation, rot2; quarter rotation, rot4), having only diagonal reflection symmetry (in one long-diagonal, dia1; in both longdiagonals, dia2), those having only orthogonal reflection symmetry (in one mid-perpendicular, ort1; in both mid-perpendiculars, ort2) and those having no symmetry (abbreviated, iden). …”

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confidence: 99%

“…Since the earlier publication [1]. I redesigned the branch-and-bound algorithm and chose another order of tree pruning.…”

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confidence: 99%

Progress in the No-Three-in-Line Problem, II

Flammenkamp

1998

Journal of Combinatorial Theory, Series A

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First solutions to the no-three-in-line problem for n=33, 37, 39, 41, 43, 45, 48, 50, 52 and for certain symmetry classes for n=26, 42, 44 are presented. All configurations with n 16 have been generated. Further, the significance of a new symmetry class for configurations which are almost as symmetric as those in class rot4 is demonstrated.1998 Academic Press, Inc.Key words and phrases: no-three-in-line; lattice configurations; branch-and-bound technique.We consider a square n_n grid in the Euclidean plane. The task is to mark as many of the intersection points as possible under the restriction that no three of the marked points lie in a straight line. One can obviously mark at most 2n points. The problem of finding for which n this value is reached is known as the no-three-in-line problem. For a short concise overview of the historical development, see [1], [2].Usually the solutions are partitioned into the following eight symmetry classes: These having full symmetry (abbreviation, full), having only rotational symmetry (half rotation, rot2; quarter rotation, rot4), having only diagonal reflection symmetry (in one long-diagonal, dia1; in both longdiagonals, dia2), those having only orthogonal reflection symmetry (in one mid-perpendicular, ort1; in both mid-perpendiculars, ort2) and those having no symmetry (abbreviated, iden).

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“…This question, dubbed the no-threein-line problem, has since been widely studied [1,2,7,14,[16][17][18][19]21]. A breakthrough came in 1951, when Erdős [14] proved that for any prime p, the set {(x, x 2 mod p) : 0 ≤ x ≤ p − 1} contains no three collinear points.…”

Section: Introductionmentioning

confidence: 99%

No-Three-in-Line-in-3D

Pór

Wood

2005

Graph Drawing

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Abstract. The no-three-in-line problem, introduced by Dudeney in 1917, asks for the maximum number of points in the n × n grid with no three points collinear. In 1951, Erdös proved that the answer is Θ(n). We consider the analogous three-dimensional problem, and prove that the maximum number of points in the n × n × n grid with no three collinear is Θ(n 2 ). This result is generalised by the notion of a 3D drawing of a graph. Here each vertex is represented by a distinct gridpoint in Z 3 , such that the line-segment representing each edge does not intersect any vertex, except for its own endpoints. Note that edges may cross. A 3D drawing of a complete graph Kn is nothing more than a set of n gridpoints with no three collinear. A slight generalisation of our first result is that the minimum volume for a 3D drawing of Kn is Θ(n 3/2 ). This compares favourably to Θ(n 3 ) when edges are not allowed to cross. Generalising the construction for Kn, we prove that every k-colourable graph on n vertices has a 3D drawing with O(n √ k) volume. For the k-partite Turán graph, we prove a lower bound of Ω ((kn) 3/4 ).

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Progress in the no-three-in-line-problem

Cited by 17 publications

References 8 publications

On the general position set of two classes of graphs

On the general position set of two classes of graphs

Progress in the No-Three-in-Line Problem, II

No-Three-in-Line-in-3D

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