Locally connected graphs: metric properties

TL;DR

This paper proves that connected locally connected graphs generally define metric spaces with at least as many lines as vertices, confirming a conjecture for this class except for three specific multipartite graphs.

Contribution

It establishes that connected locally connected graphs mostly satisfy a conjecture relating the number of lines in their metric space to the number of vertices, with only three exceptions.

Findings

Connected locally connected graphs define metric spaces with at least as many lines as vertices.

The conjecture holds for all such graphs except three specific multipartite cases.

The work confirms a broader conjecture in metric space theory.

Abstract

In this work we show that any connected locally connected graph defines a metric space having at least as many lines as vertices with only three exception: the complete multipartite graphs $K_{1,2,2}$, $K_{2,2,2}$ and $K_{2,2,2,2}$. This proves that this class fulfills a conjecture, proposed by Chen and Chv\'atal, saying that any metric space on n points has at least n lines or a line containing all the points.