2025/11/16 by Xu, Songnian, Wong, Dein, Zhen, Wenhao
#05C25 #FOS: Mathematics #Group Theory (math.GR)
paper · doi:10.48550/arxiv.2511.12660
Let G be a finite group and m ≥ 2 a positive integer. We say that G admits an oriented m-semiregular representation (abbreviated as OmSR) if there exists a m-Cayley digraph Γ over G such that Γ is oriented and Aut(Γ) ≅ G. In \citexu1, we classified finite groups generated by at most two elements that admit an OmSR of valency 3 for m ≥ 2 and G \ncong ℤ1. In this article, we consider m-partite digraphs.We say a finite group G admits an m-partite oriented semiregular representation (m-partite digraphical representation), abbreviated as m-POSR (m-PDR), if there exists an oriented m-partite Cayley digraph (m-partite Cayley digraph) Γ with Aut(Γ) ≅ G. In this paper, we classify finite groups generated by at most two elements that admit m-POSR. Since if G admits an m-POSR, then G must also admit an m-PDR (while the converse does not hold), as a natural consequence, we also provide a complete classification for groups G=⟨ x,y⟩ that admit m-PDR of valency 3. This complements the results in \citexu2.