A De Bruijn-Erdős Theorem in Graphs?

Abstract

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.

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

“…The result presented in this article solves Problem 3 of Chvátal's survey [8]. It would interesting to extends this result to prove that Conjecture 2.3 in [3] holds for the class of HH‐free graphs.…”

“…It actually answers Problem 3 of Chvátal's survey [8]. From now on, we let $${\rm{ {\mathcal H} }}$$ denote the class of {house, hole}‐free graphs.…”

“…Kantor on her own [11] and together with Patkós [12] have been investigating the conjecture in the Euclidean plane equipped with $${L}_{1}$$‐metric. For a good overview of previous results and open problems, one may read the enjoyable survey written by Chvátal in 2018 [8].…”

Metric hypergraphs and metric lines

“…Early progress toward the conjecture that this generalization does hold true is surveyed in [6]; contributions too recent to be included there are [3], [4], [8], [11], [12].…”

Chen and Chvátal's Conjecture in tournaments

“…It has also been solved for graphs of diameter at most two but requiring much more effort. To the best of our knowledge it is still open for graphs of diameter at most three (see [8] for a recent survey).…”