Locally connected graphs: metric properties

TL;DR

This paper analyzes metric properties induced by connected locally connected graphs, focusing on the number of lines defined by vertex pairs. It develops a diameter-based counting framework, introducing $\mathcal{L}^z$ and a betweenness-based analysis, with key lemmas that connect line equality to betweenness in $\mathcal{B}_G$. The main result establishes that any $G\in \ell\mathcal{C}$ on $n$ vertices has at least $n$ lines, except for the complete multipartite graphs $K_{1,2,2}$, $K_{2,2,2}$, and $K_{2,2,2,2}$, which have fewer, thereby proving the Chen-Chvátal conjecture for this graph class. The work also discusses limitations, providing counterexamples to broader extensions and deriving corollaries for certain planar graphs, highlighting both the scope and boundaries of the method.

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 Chvátal, saying that any metric space on n points has at least n lines or a line containing all the points.

Theorems & Definitions (35)

• Proposition 1
• proof
• Theorem 2
• Conjecture 1
• Theorem 3: Theorem 2.1 in abmaroza
• Lemma 4
• proof
• Lemma 5
• proof
• Proposition 6