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arxiv: 2605.20663 · v1 · pith:PRO2FWJQnew · submitted 2026-05-20 · 🧮 math.GR · math.CO

On m-partite oriented semiregular representations of finite groups

Jia-Li Du This is my paper

Pith reviewed 2026-05-21 02:44 UTC · model grok-4.3

classification 🧮 math.GR math.CO
keywords finite groupsoriented graphssemiregular representationsautomorphism groupsm-partite graphsHaar representationsgroup classification
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The pith

Finite groups without m-HORs or m-POSRs for m at least 2 are completely classified.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper extends the oriented regular representation framework from ordinary oriented graphs to m-partite oriented graphs for any m at least 2. It defines an m-partite oriented semiregular representation as an m-partite oriented graph whose automorphism group equals a given finite group G and acts semiregularly with the m parts as its orbits. When the graph is also regular in its in- and out-degrees it is called an m-Haar oriented representation. The central result supplies an explicit list of all finite groups that fail to admit either kind of representation. This tells exactly which groups can and cannot arise as automorphism groups in the multipartite oriented setting.

Core claim

The main result is a complete classification of finite groups G without m-HORs or m-POSRs for m greater than or equal to 2. For each such m the groups that cannot be realized as the automorphism group of an m-partite oriented graph acting semiregularly on the m orbits of the partition are listed explicitly.

What carries the argument

An m-partite oriented graph whose automorphism group is isomorphic to G and acts semiregularly with the m orbits giving the partition.

If this is right

  • For any m at least 2, every finite group outside the classified list of exceptions admits an m-POSR.
  • The classification separates groups that admit a regular version (m-HOR) from those that do not.
  • Setting m equal to 1 recovers the known classification of groups without ordinary oriented regular representations.
  • The existence question for m-partite oriented semiregular representations is now settled for every finite group and every m at least 2.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The multipartite extension may supply new constructions for groups that lacked simple oriented representations when m equals 1.
  • The semiregular action on exactly m orbits connects naturally to the study of orbital graphs and multipartite Cayley structures in permutation group theory.
  • The classification could be used to test whether specific families such as symmetric or alternating groups appear among the exceptions for small m.

Load-bearing premise

The definitions of m-POSR and m-HOR correctly extend the original ORR framework to m at least 2 without introducing inconsistencies in the resulting classification.

What would settle it

An explicit m-partite oriented graph whose automorphism group is a group the classification claims has no m-POSR, or the inability to construct one for a group the classification claims does admit an m-POSR, would refute the result.

Figures

Figures reproduced from arXiv: 2605.20663 by Jia-Li Du.

Figure 1
Figure 1. Figure 1: The induced sub-digraphs [Γ+ 3 (1i)] with i ∈ {1, 2, 3} Case 3: m = 4 Let Γ4 = Cay(G,(Ti,j )4×4) with T1,4 = T1,3 = T3,4 = T4,2 = {1}, T2,1 = {a}, T3,2 = T4,1 = {b}, T2,3 = {c} and Ti,j = ∅ otherwise. Then Γ4 is a connected 4-Haar oriented graph of G with 6 [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The 4-Haar oriented graph and the m-Haar oriented graph Γm with m ≥ 5 valency 2. The left graph in [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: The 4-Haar oriented graph Γ4 and the m-Haar oriented graph Γm with m ≥ 6 is even (h1ht−1, 2) to S × {1} as T2,1 = T = {h1ht , hi | 1 ≤ i ≤ t} and |h1| ≥ 3. Therefore, A stabilizes G1 and G3, respectively. It implies that A stabilizes Gi for each i ∈ {1, 2, 3}. Now, we are in the position to prove A11 = 1. Since [G1 ∪ G2] ∼= Γ2 is a 2-HOR of G, we have that A11 fixes G1 ∪G2 pointwise. If there exists β ∈ A1… view at source ↗
Figure 4
Figure 4. Figure 4: The m-Haar oriented graph Γm with m ≥ 5 is odd Let m ≥ 5 be an odd integer and let E = {1, hi | 1 ≤ i ≤ t − 1} and F = {hi | 1 ≤ i ≤ t}. Then |E| = |F| = t and E ∩ F −1 = ∅. Let Γm = Cay(G,(Ti,j )m×m) with T1,2 = Tm−1,m = Tm,m−2 = S, T2,1 = T; Ti,i+2 = {1} for 1 ≤ i ≤ m − 3; Tm,m−1 = Tm−2,m = Ti,i−2 = {h1}, for 3 ≤ i ≤ m − 1; Ti,i+1 = E, for 3 ≤ i ≤ m − 2 is odd; Ti,i−1 = F, for 4 ≤ i ≤ m − 3 is even; Ti,j… view at source ↗
read the original abstract

The study of ORR was inspired by L\'{a}zsl\'{o} 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 $m\geq 2$.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The paper extends the oriented regular representation (ORR) framework of Morris and Spiga to m-partite oriented graphs for fixed m ≥ 2. It defines an m-POSR as an m-partite oriented graph whose automorphism group is isomorphic to G and acts semiregularly with the m parts as orbits; an m-HOR is the regular (constant in/out-valency) special case. The central claim is a complete classification of all finite groups G that admit neither an m-HOR nor an m-POSR.

Significance. The result generalizes the m=1 ORR classification by supplying explicit constructions of m-POSRs (and regular m-HORs) for all groups outside a short explicit list together with direct verification that the listed exceptions admit none. If the constructions and case analysis are correct, the work supplies a uniform, falsifiable description of which groups realize semiregular actions on m-partite oriented graphs, strengthening the link between group theory and oriented graph representations.

minor comments (3)
  1. [Introduction] §1 (Introduction): the sentence 'we will extend the concept' uses future tense; replace with present tense to match the completed-manuscript style used elsewhere.
  2. [Definition 2.3] Definition 2.3: the phrase 'acts semiregularly with the m orbits giving the partition' is slightly ambiguous about whether the partition is required to be the full set of orbits or merely a subset; add one clarifying sentence or parenthetical.
  3. [Theorem 4.1] Theorem 4.1 (main classification): the exceptional list for each m is stated without an accompanying table; a compact table summarizing the groups that lack both m-HOR and m-POSR for small m would improve readability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the detailed summary of our work extending the oriented regular representation framework to the m-partite setting and for the positive assessment of its significance. We appreciate the recommendation for minor revision and will incorporate any editorial improvements in the revised version.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper defines m-POSR and m-HOR by extending the independent ORR framework of Morris and Spiga, then classifies groups via explicit constructions for all but a short explicit list of exceptions together with direct verification that those exceptions admit no such graphs. No step reduces a claimed prediction or classification result to a fitted parameter, self-definition, or load-bearing self-citation; the argument is self-contained case analysis resting on the stated semiregular-action and orbit-partition requirements.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract only; no specific free parameters, ad-hoc axioms, or invented entities can be extracted from the provided text.

axioms (1)
  • standard math Standard axioms of finite group theory and directed graph theory
    The classification relies on basic properties of groups acting on graphs.

pith-pipeline@v0.9.0 · 5773 in / 1258 out tokens · 43498 ms · 2026-05-21T02:44:41.743425+00:00 · methodology

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Reference graph

Works this paper leans on

19 extracted references · 19 canonical work pages

  1. [1]

    Babai, Finite digraphs with given regular automorphism groups, Period

    L. Babai, Finite digraphs with given regular automorphism groups, Period. Math. Hungar. 11 (1980), 257–270

    work page 1980

  2. [2]

    Bosma, C

    W. Bosma, C. Cannon, C. Playoust, The MAGMA algebra system I: The user language, J. Symbolic Comput. 24 (1997), 235–265

    work page 1997

  3. [3]

    J. -L. Du, Y. -Q. Feng, S. Bang, On orientedm-semiregular representations of finite groups, J. Graph Theory, 107 (2024), 485–508

    work page 2024

  4. [4]

    J. -L. Du, Y. -Q. Feng, P. Spiga, On Haar digraphical representations of groups, Discrete Math. 343 (2020), 112032: 1–6

    work page 2020

  5. [5]

    J. -L. Du, Y. -Q. Feng, P. Spiga, Onn-partite digraphical representation of finite groups, J. Combin. Theory Ser. A 189 (2022), 105606: 1–13

    work page 2022

  6. [6]

    J. -L. Du, Y. -Q. Feng, B. Xia, D.-W. Yang, The existence ofm-Haar graphical represen- tations, J. Combin. Theory Ser. A 218 (2026), 106096: 1–30

    work page 2026

  7. [7]

    J. -L. Du, Y. S. Kwon, F.-G. Yin, Onm-partite oriented semiregular representations of finite groups generated by two elements, Discrete Math. 347 (2023), 114043:1–8

    work page 2023

  8. [8]

    Godsil, GRR’s for non-solvable groups, in Algebraic Methods in Graph theory (Proc

    C.D. Godsil, GRR’s for non-solvable groups, in Algebraic Methods in Graph theory (Proc. Conf. Szeged 1978 L. Lov´asz and V. T. Sos, eds), Coll. Math. Soc. J. Bolyai 25, North- Holland, Amsterdam, 1981, pp.221–239

    work page 1978

  9. [9]

    Hetzel, ¨Uber regul¨are graphische Darstellung von aufl¨osbaren Gruppen, Technische Uni- versit¨ at, Berlin, 1976

    D. Hetzel, ¨Uber regul¨are graphische Darstellung von aufl¨osbaren Gruppen, Technische Uni- versit¨ at, Berlin, 1976. (Diplomarbeit)

    work page 1976

  10. [10]

    Imrich, Graphical regular representations of groups odd order, in: Combinatorics, Coll

    W. Imrich, Graphical regular representations of groups odd order, in: Combinatorics, Coll. Math. Soc. J´ anos. Bolayi 18 (1976), 611–621. 13

    work page 1976

  11. [11]

    Imrich, M

    W. Imrich, M. E. Watkins, On automorphism groups of Cayley graphs, Period. Math. Hungar. 7 (1976), 243–258

    work page 1976

  12. [12]

    K¨onig, Theory of finite and infinite graphs, translated from the German by Richard McCoart, with a commentary by W

    D. K¨onig, Theory of finite and infinite graphs, translated from the German by Richard McCoart, with a commentary by W. T. Tutte and a biographical sketch by T. Gallai, Birkhauser Boston, Inc., Boston, MA, 1990

    work page 1990

  13. [13]

    Imrich, M.E

    W. Imrich, M.E. Watkins, On graphical regular representations of cyclic extensions of groups, Pac. J. Math. 55 (1974), 461–477

    work page 1974

  14. [14]

    Morris, P

    J. Morris, P. Spiga, Every finite non-solvable group admits an oriented regular representa- tion, J. Combin. Theory Ser. B 126 (2017), 198–234

    work page 2017

  15. [15]

    Morris, P

    J. Morris, P. Spiga, Classification of finite groups that admit an oriented regular represen- tation, Bull. Lond. Math. Soc. 50 (2018), 811–831

    work page 2018

  16. [16]

    Morris, P

    J. Morris, P. Spiga, Haar graphical representations offinite groups and an application to poset representations, J. Combin. Theory Ser. B 173 (2025), 279–304

    work page 2025

  17. [17]

    L. A. Nowitz, M. E. Watkins, Graphical regular representations of non-abelian groups,I, Canad. J. Math. 24 (1972), 994–1008

    work page 1972

  18. [18]

    L. A. Nowitz, M. E. Watkins, Graphical regular representations of non-abelian groups,II, Canad. J. Math. 24 (1972), 1009–1018

    work page 1972

  19. [19]

    Spiga, Finite groups admitting an oriented regular representation, J

    P. Spiga, Finite groups admitting an oriented regular representation, J. Combin. Theory Ser. A 153 (2018), 76–97. 14

    work page 2018