A digraph version of the Friendship Theorem
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2023
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| _version_ | 1866913945098387456 |
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| author | Choi, Myungho Chu, Hojin Kim, Suh-Ryung |
| author_facet | Choi, Myungho Chu, Hojin Kim, Suh-Ryung |
| contents | The Friendship Theorem states that if in a party any pair of persons has precisely one common friend, then there is always a person who is everybody's friend and the theorem has been proved by Paul Erdős, Alfréd Rényi, and Vera T. Sós in 1966. ``What would happen if instead any pair of persons likes precisely one person?" While a friendship relation is symmetric, a liking relation may not be symmetric. Therefore to represent a liking relation we should use a directed graph. We call this digraph a ``liking digraph". It is easy to check that a symmetric liking digraph becomes a friendship graph if each directed cycle of length two is replaced with an edge. In this paper, we provide a digraph formulation of the Friendship Theorem which characterizes the liking digraphs. We also establish a sufficient and necessary condition for the existence of liking digraphs. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2305_04058 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | A digraph version of the Friendship Theorem Choi, Myungho Chu, Hojin Kim, Suh-Ryung Combinatorics 05C20, 05C75 The Friendship Theorem states that if in a party any pair of persons has precisely one common friend, then there is always a person who is everybody's friend and the theorem has been proved by Paul Erdős, Alfréd Rényi, and Vera T. Sós in 1966. ``What would happen if instead any pair of persons likes precisely one person?" While a friendship relation is symmetric, a liking relation may not be symmetric. Therefore to represent a liking relation we should use a directed graph. We call this digraph a ``liking digraph". It is easy to check that a symmetric liking digraph becomes a friendship graph if each directed cycle of length two is replaced with an edge. In this paper, we provide a digraph formulation of the Friendship Theorem which characterizes the liking digraphs. We also establish a sufficient and necessary condition for the existence of liking digraphs. |
| title | A digraph version of the Friendship Theorem |
| topic | Combinatorics 05C20, 05C75 |
| url | https://arxiv.org/abs/2305.04058 |