A digraph version of the Friendship Theorem
Abstract
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 Erd\H{o}s, Alfr\'{e}d R\'{e}nyi, and Vera T. S\'{o}s 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.
Cite
@article{arxiv.2305.04058,
title = {A digraph version of the Friendship Theorem},
author = {Myungho Choi and Hojin Chu and Suh-Ryung Kim},
journal= {arXiv preprint arXiv:2305.04058},
year = {2025}
}