Finite groups admitting a regular tournament $m$-semiregular representation

TL;DR

This work resolves the existence of regular tournament $m$-semiregular representations (TmSR) for finite groups of odd order by constructing regular $m$-Cayley digraphs $\Gamma={\rm Cay}(G, T_{i,j})$ with $m$ disjoint $G$-orbits and proving ${\rm Aut}(\Gamma)={\rm Reg}(G)$ for all odd $m\ge3$ when $|G|=n>1$. The authors organize the proof into three cases corresponding to parameter ranges and group structure: Case I ($m\ge7$ or $m=5$, $n\ge5$), Case II ($m=5$, $n=3$), and Case III ($m=3$), handling both the presence/absence of a tournament regular representation (TRR) and the exceptional $Z_3^2$ and $Z_3^3$ cases with explicit constructions. The main result is that every nontrivial finite group of odd order admits a regular TmSR for every odd $m\ge3$, extending prior TRR results and yielding explicit, constructive representations via Cayley digraphs. The work thereby completes the classification for odd-order groups in this regular-tournament semiregular setting, with the trivial group remaining a singular exception.

Abstract

For a positive integer $m$, a finite group $G$ is said to admit a tournament $m$-semiregular representation (TmSR for short) if there exists a tournament $Γ$ such that the automorphism group of $Γ$ is isomorphic to $G$ and acts semiregularly on the vertex set of $Γ$ with $m$ orbits. Clearly, every finite group of even order does not admit a TmSR for any positive integer $m$, and T1SR is the well-known tournament regular representation (TRR for short). In 1986, Godsil \cite{god} proved, by a probabilistic approach, that the only finite groups of odd order without a TRR are $\mathbb{Z}_3^2$ and $\mathbb{Z}_3^3$ . More recently, Du \cite{du} proved that every finite group of odd order has a TmSR for every $m \geq 2$. The author of \cite{du} observed that a finite group of odd order has no regular TmSR when $m$ is an even integer, a group of order $1$ has no regular T3SR, and $\mathbb{Z}_3^2$ admits a regular T3SR. At the end of \cite{du}, Du proposed the following problem. \noindent{\sf\it Problem.} \ \ {\it For every odd integer $m\geq 3$, classify finite groups of odd order which have a regular TmSR.} The motivation of this paper is to give an answer for the above problem. We proved that if $G$ is a finite group with odd order $n>1$, then $G$ admits a regular TmSR for any odd integer $m\geq 3$.

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