Digraphs in which every $t$ vertices have exactly $λ$ common out-neighbors
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arXiv
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| Format: | Preprint |
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2024
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| author | Choi, Myungho Chu, Hojin Kim, Suh-Ryung |
| author_facet | Choi, Myungho Chu, Hojin Kim, Suh-Ryung |
| contents | We say that a digraph is a $(t,λ)$-liking digraph if every $t$ vertices have exactly $λ$ common out-neighbors. In 1975, Plesník [Graphs with a homogeneity, 1975. {\it Glasnik Mathematicki} 10:9-23] proved that any $(t,1)$-liking digraph is the complete digraph on $t+1$ vertices for each $t\geq 3$. Choi {\it et al}. [A digraph version of the Friendship Theorem, 2025. {\it Discrete mathematics}, 348(1), 114238] showed that a $(2,1)$-liking digraph is a fancy wheel digraph or a $k$-diregular digraph for some positive integer $k$. In this paper, we extend these results by completely characterizing the $(t,λ)$-liking digraphs with $t \geq λ+2$ and giving some equivalent conditions for a $(t,λ)$-liking digraph being a complete digraph on $t+λ$ vertices. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_02662 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Digraphs in which every $t$ vertices have exactly $λ$ common out-neighbors Choi, Myungho Chu, Hojin Kim, Suh-Ryung Combinatorics 05C20, 05C75 We say that a digraph is a $(t,λ)$-liking digraph if every $t$ vertices have exactly $λ$ common out-neighbors. In 1975, Plesník [Graphs with a homogeneity, 1975. {\it Glasnik Mathematicki} 10:9-23] proved that any $(t,1)$-liking digraph is the complete digraph on $t+1$ vertices for each $t\geq 3$. Choi {\it et al}. [A digraph version of the Friendship Theorem, 2025. {\it Discrete mathematics}, 348(1), 114238] showed that a $(2,1)$-liking digraph is a fancy wheel digraph or a $k$-diregular digraph for some positive integer $k$. In this paper, we extend these results by completely characterizing the $(t,λ)$-liking digraphs with $t \geq λ+2$ and giving some equivalent conditions for a $(t,λ)$-liking digraph being a complete digraph on $t+λ$ vertices. |
| title | Digraphs in which every $t$ vertices have exactly $λ$ common out-neighbors |
| topic | Combinatorics 05C20, 05C75 |
| url | https://arxiv.org/abs/2405.02662 |