Chen-Chv\'atal Conjecture for Graphs of Diameter 3

TL;DR

This paper proves that all graphs with diameter three satisfy the Chen-Chvátal conjecture, which states that such metric spaces have at least as many lines as points, extending a classical geometric result.

Contribution

The paper characterizes all diameter-three graphs with fewer lines than points and confirms the Chen-Chvátal conjecture for these graphs.

Findings

Identified all diameter-three graphs with fewer lines than vertices.

Proved that graphs of diameter three satisfy the Chen-Chvátal conjecture.

Extended the conjecture's validity to a new class of graphs.

Abstract

In 2008, Chen and Chv\'atal conjectured that in every finite metric space of $n$ points, there are at least $n$ distinct lines, or the whole set of points is a line. This is a generalization of a classical result in the Euclidean plane. The Chen-Chv\'atal conjecture is open even in metric spaces induced by connected graphs. In 2018, it was asked by Chv\'atal whether graphs of diameter three satisfy the conjecture. In this work, we find all graphs of diameter three having fewer lines than vertices. As a direct consequence, we prove that graphs of diameter three satisfy the Chen-Chv\'atal conjecture.