Lines on digraphs of low diameter

TL;DR

This paper investigates the number of lines in metric spaces derived from directed graphs with low diameter, extending previous conjectures and proving new results for specific classes of digraphs.

Contribution

It extends the conjecture about lines in metric spaces to quasi-metric spaces from digraphs of low diameter and proves it for several classes of bipartite and oriented graphs.

Findings

Validates the conjecture for bipartite digraphs of diameter ≤ 3

Proves the conjecture for oriented graphs of diameter 2

Establishes the conjecture for digraphs of diameter 3 and girth 4

Abstract

A set of n non-collinear points in the Euclidean plane defines at least n different lines. Chen and Chv\'tal in 2008 conjectured that the same results is true in metric spaces for an adequate definition of line. More recently, it was conjectured in 2018 by Aboulker et al. that any large enough bridgeless graph on n vertices defines a metric space that has at least n lines. We study the natural extension of Aboulker et al.'s conjecture into the context of quasi-metric spaces defined by digraphs of low diameter. We prove that it is valid for quasi-metric spaces defined by bipartite digraphs of diameter at most three, oriented graphs of diameter two and, digraphs of diameter three and directed girth four.