Graphs with no induced house nor induced hole have the de\n Bruijn-Erd Hos property

2020/05/19 by P Aboulker, Aboulker, Pierre, Laurent Beaudou +5 · 2 citations
Computer Science · Mathematics · #Advanced Combinatorial Mathematics #Advanced Graph Theory Research #Combinatorics (math.CO) #Discrete Mathematics (cs.DM) #FOS: Computer and information sciences #FOS: Mathematics #Limits and Structures in Graph Theory

paper · pdf · doi:10.48550/arxiv.2005.09447

openalex publication_date 2020/05/19 · openalex created_date 2022/09/15 · openalex updated_date 2026/07/28

Abstract

A set of n points in the plane which are not all collinear defines at least n\ndistinct lines. Chen and Chv 'atal conjectured in 2008 that a similar result\ncan be achieved in the broader context of finite metric spaces. This conjecture\nremains open even for graph metrics. In this article we prove that graphs with\nno induced house nor induced cycle of length at least~5 verify the desired\nproperty. We focus on lines generated by vertices at distance at most 2, define\na new notion of ``good pairs'' that might have application in larger families,\nand finally use a discharging technique to count lines in irreducible graphs.\n

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• Chen-Chvátal Conjecture for Graphs of Diameter 3
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