A characterization of graphs of diameter two with fewer lines than vertices
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arXiv
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866912664930746368 |
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| author | Matamala, Martín |
| author_facet | Matamala, Martín |
| contents | In 2008 Chen and Chvátal conjectured that any metric space on n points has at least n lines, unless all the points belong to one line. Chv\atal proved in 2014 that this is indeed the case for metric spaces with distances 0, 1 and 2. In this work, we prove that there exists a family of ten graphs such that a metric space defined by a graph of diameter two has fewer lines than points if and only if the associated graph belongs to that family. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_19601 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A characterization of graphs of diameter two with fewer lines than vertices Matamala, Martín Combinatorics Metric Geometry 05C62, 05c12, 54E35 In 2008 Chen and Chvátal conjectured that any metric space on n points has at least n lines, unless all the points belong to one line. Chv\atal proved in 2014 that this is indeed the case for metric spaces with distances 0, 1 and 2. In this work, we prove that there exists a family of ten graphs such that a metric space defined by a graph of diameter two has fewer lines than points if and only if the associated graph belongs to that family. |
| title | A characterization of graphs of diameter two with fewer lines than vertices |
| topic | Combinatorics Metric Geometry 05C62, 05c12, 54E35 |
| url | https://arxiv.org/abs/2510.19601 |