A New Class of Graphs That Satisfies the Chen‐Chvátal Conjecture

Aboulker, Pierre; Matamala, Martı́n; Rochet, Paul; Zamora, José

doi:10.1002/jgt.22142

Cited by 15 publications

(24 citation statements)

References 17 publications

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“…A stronger conjecture interpolates between the two operands of the disjunction: Theorem 2.1 of [5] is stronger than Theorem 15: except for six graphs that have the AMRZ property, it replaces the AMRZ property by the property that the number of lines plus the number of bridges is at least the number of vertices.…”

Section: An Interpolationmentioning

confidence: 99%

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A De Bruijn-Erdős Theorem in Graphs?

Chvátal

2018

Graph Theory

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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: An Interpolationmentioning

confidence: 99%

“…[5, Conjecture 2.2] In all graphs with no pendant edges except for finitely many cases, the number of lines plus the number of bridges is at least the number of vertices.Here are Problems 3, 4, 5, 6 with 'DBE property' replaced by 'AMRZ property' and phrased more cautiously:Problem 14. True or false?…”

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confidence: 99%

A De Bruijn-Erdős Theorem in Graphs?

Chvátal

2018

Graph Theory

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“…Theorem 2 (Theorem 2.1 in Aboulker et al [3]). Every graph G such that every induced subgraph of G is either a chordal graph, has a cut vertex or a nontrivial module satisfies G G G ( ) + BR( ) | | ℓ ≥ , unless G is one of the six graphs depicted in Figure 1.…”

Section: Introductionmentioning

confidence: 95%

“…On the other hand, an edge e of a graph G is a bridge if G e − has more connected components than the graph G. Let G BR( ) denote the number of bridges of G. Notice that if ab is a bridge of G, then the line ab G is universal. The main result in [3] is the following, where G | | denotes the number of vertices of G.…”

Section: Introductionmentioning

confidence: 99%

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Lines in bipartite graphs and in 2‐metric spaces

Matamala

Zamora

2020

Journal of Graph Theory

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The line generated by two distinct points, x and y, in a finite metric space M V d = (,), is the set of points given by z V d x y d x z d z y d x y d x z d z y { : (,) = | (,) + (,)|or (,) = | (,) − (,)|}. ∈ It is denoted by xy M. A 2-set x y { , } such that xy V = M is called a universal pair and its generated line a universal line. Chen and Chvátal conjectured that in any finite metric space either there is a universal line, or there are at least |V| different (nonuniversal) lines. Chvátal proved that this is indeed the case when the metric space has distances in the set {0, 1, 2}. Aboulker et al proposed the following strengthenings for Chen and Chvátal conjecture in the context of metric spaces induced by finite graphs: First, the number of lines plus the number of bridges of the graph is at least the number of points. Second, the number of lines plus the number of universal pairs is at least the number of points of the space. In this study, we prove that the first conjecture is true for bipartite graphs different from C 4 or K 2,3 , and that the second conjecture is true for metric spaces with distances in the set {0, 1, 2}.

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Graphs with no induced house nor induced hole have the de Bruijn–Erdös property

Aboulker

Beaudou

Matamala

et al. 2022

Journal of Graph Theory

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A set of n $n$ points in the plane which are not all collinear defines at least n $n$ distinct lines. Chen and Chvátal conjectured in 2008 that a similar result can be achieved in the broader context of finite metric spaces. This conjecture remains open even for graph metrics. In this article we prove that graphs with no induced house nor induced cycle of length at least 5 verify the desired property. We focus on lines generated by vertices at distance at most 2, define a new notion of 'good pairs' that might have application in larger families, and finally use a discharging technique to count lines in irreducible graphs.

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A New Class of Graphs That Satisfies the Chen‐Chvátal Conjecture

Cited by 15 publications

References 17 publications

A De Bruijn-Erdős Theorem in Graphs?

A De Bruijn-Erdős Theorem in Graphs?

Lines in bipartite graphs and in 2‐metric spaces

Graphs with no induced house nor induced hole have the de Bruijn–Erdös property

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