Structural and Complexity Aspects of Line Systems of Graphs

Jirásek, Jozef; Klavík, Pavel

doi:10.1007/978-3-642-17517-6_16

Cited by 2 publications

(2 citation statements)

References 15 publications

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“…• In August 2010, Google Scholar surprised me by telling me that we were not alone in the universe: others were interested in Conjecture 1, too. I cannot resist the temptation to quote the opening of [27] verbatim (except for the reference labels):…”

Section: A Logbookmentioning

confidence: 99%

A De Bruijn-Erdős Theorem in Graphs?

Chvátal

2018

Problem Books in Mathematics

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A set of n points in the Euclidean plane determines at least n distinct lines unless these n points are collinear. In 2006, Chen and Chvátal asked whether the same statement holds true in general metric spaces, where the line determined by points x and y is defined as the set consisting of x, y, and all points z such that one of the three points x, y, z lies between the other two. The conjecture that it does hold true remains unresolved even in the special case where the metric space arises from a connected undirected graph with unit lengths assigned to edges. We trace its curriculum vitae and point out twenty-nine related open problems plus three additional conjectures.

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Section: A Logbookmentioning

confidence: 99%

A De Bruijn-Erdős Theorem in Graphs?

Chvátal

2018

Problem Books in Mathematics

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“…In [2], the same authors generalize the de Bruijn-Erdős theorem in a different direction. Several results dealing with lines in metric spaces induced by graphs were also proved by Jirásek and Klavík in [11].…”

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confidence: 81%

Towards a de Bruijn–Erdős Theorem in the $$L_1$$ -Metric

Kantor

Patkós

2013

Discrete Comput Geom

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A well-known theorem of de Bruijn and Erdős states that any set of n non-collinear points in the plane determines at least n lines. Chen and Chvátal asked whether an analogous statement holds within the framework of finite metric spaces, with lines defined using the notion of betweenness.In this paper, we prove that the answer is affirmative for sets of n points in the plane with the L 1 metric, provided that no two points share their x-or y-coordinate. In this case, either there is a line that contains all n points, or X induces at least n distinct lines.If points of X are allowed to share their coordinates, then either there is a line that contains all n points, or X induces at least n/37 distinct lines. Lines in metric spacesTwo well-known results are known under the name "de Bruijn-Erdős theorem". One of them, published in [8] in 1948, states that every set of n points in the plane is either collinear or it determines at least n distinct lines.The notion of a line can be extended naturally into an arbitrary metric space. If (V, ρ) is an arbitrary metric space and a, b, x ∈ V , we say that x is between the points a and b if ρ(a, b) = ρ(a, x) + ρ(x, b).

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Structural and Complexity Aspects of Line Systems of Graphs

Cited by 2 publications

References 15 publications

A De Bruijn-Erdős Theorem in Graphs?

A De Bruijn-Erdős Theorem in Graphs?

Towards a de Bruijn–Erdős Theorem in the $$L_1$$ -Metric

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