On $m$-partite oriented semiregular representations of finite groups
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866911700402307072 |
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| author | Du, Jia-Li |
| author_facet | Du, Jia-Li |
| contents | The study of ORR was inspired by Lázsló Babai in 1980 when he asked a question: Which [finite] groups admit an oriented graph as a DRR? And it has been solved by Joy Morris and Pablo Spiga through a series of papers in 2018. In this paper, we will extend the concept of ORR
to $m$-partite oriented graphs for $m\geq 2$. We say that a finite group $G$ admits an \emph{$m$-partite oriented semiregular representation} ($m$-POSR) if there exists an $m$-partite oriented graph $\G$ such that its automorphism group is isomorphic to $G$ and acts semiregularly with the $m$ orbits giving the partition.
Moreover, if $\G$ is regular,
that is, each vertex has
the same in- and out-valency,
it can be viewed as the oriented version of an $m$-Haar graph of $G$ and we call $\G$ is an \emph{$m$-Haar oriented representation} ($m$-HOR) of $G$.
Our main result is a complete classification of finite groups $G$ without $m$-HORs or $m$-POSRs for $m\geq 2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_20663 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On $m$-partite oriented semiregular representations of finite groups Du, Jia-Li Group Theory Combinatorics 05C25, 20B25 The study of ORR was inspired by Lázsló Babai in 1980 when he asked a question: Which [finite] groups admit an oriented graph as a DRR? And it has been solved by Joy Morris and Pablo Spiga through a series of papers in 2018. In this paper, we will extend the concept of ORR to $m$-partite oriented graphs for $m\geq 2$. We say that a finite group $G$ admits an \emph{$m$-partite oriented semiregular representation} ($m$-POSR) if there exists an $m$-partite oriented graph $\G$ such that its automorphism group is isomorphic to $G$ and acts semiregularly with the $m$ orbits giving the partition. Moreover, if $\G$ is regular, that is, each vertex has the same in- and out-valency, it can be viewed as the oriented version of an $m$-Haar graph of $G$ and we call $\G$ is an \emph{$m$-Haar oriented representation} ($m$-HOR) of $G$. Our main result is a complete classification of finite groups $G$ without $m$-HORs or $m$-POSRs for $m\geq 2$. |
| title | On $m$-partite oriented semiregular representations of finite groups |
| topic | Group Theory Combinatorics 05C25, 20B25 |
| url | https://arxiv.org/abs/2605.20663 |