Graphs with no induced house nor induced hole have the de Bruijn-Erd\H{o}s property

TL;DR

This paper proves that certain graphs without specific induced subgraphs satisfy a property related to the de Bruijn-Erdős conjecture, using combinatorial and discharging techniques.

Contribution

It establishes the de Bruijn-Erdős property for graphs excluding induced houses and large cycles, introducing new concepts like 'good pairs' and applying discharging methods.

Findings

Graphs with no induced house or cycle of length ≥5 satisfy the property.

Introduction of 'good pairs' for analyzing line generation.

Use of discharging technique to count lines in irreducible graphs.

Abstract

A set of n points in the plane which are not all collinear defines at least n distinct lines. Chen and Chv\'atal conjectured in 2008 that a similar result can be achieved in the broader context of finite metric spaces. This conjecture remains open even for graph metrics. In this article we prove that graphs with no induced house nor induced cycle of length at least~5 verify the desired property. We focus on lines generated by vertices at distance at most 2, define a new notion of ``good pairs'' that might have application in larger families, and finally use a discharging technique to count lines in irreducible graphs.