Lines in bipartite graphs and in 2‐metric spaces

Abstract

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}.

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

“…More generally, any graph metric defined by a graph $$G$$ such that every induced subgraph of $$G$$ is either a chordal graph, has a cut‐vertex or a nontrivial module [3]. Several strengthenings of the initial conjecture have been suggested [13]. Kantor on her own [11] and together with Patkós [12] have been investigating the conjecture in the Euclidean plane equipped with $${L}_{1}$$‐metric.…”

Graphs with no induced house nor induced hole have the de Bruijn-Erdős property

“…More generally, any graph metric defined by a graph G such that every induced subgraph of G is either a chordal graph, has a cut-vertex or a non-trivial module [3]. Several strengthenings of the initial conjecture have been suggested [11]. For a good overview of previous results and open problems, one may read the enjoyable survey written by Chvátal in 2018 [8].…”