ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 23 (2023) #P2.10
https://doi.org/10.26493/1855-3974.2857.07b
(Also available at http://amc-journal.eu)
Locally s-arc-transitive graphs arising from
product action*
Michael Giudici
Department of Mathematics and Statistics, The University of Western Australia,
Perth WA 6009, Australia
Eric Swartz †
Department of Mathematics, William & Mary,
P.O. Box 8795, Williamsburg, VA 23187-8795, USA
Received 30 March 2022, accepted 14 September 2022, published online 13 December 2022
Abstract
We study locally s-arc-transitive graphs arising from the quasiprimitive product action
(PA). We prove that, for any locally (G, 2)-arc-transitive graph with G acting quasiprimi-
tively with type PA on both G-orbits of vertices, the group G does not act primitively on
either orbit. Moreover, we construct the first examples of locally s-arc-transitive graphs of
PA type that are not standard double covers of s-arc-transitive graphs of PA type, answering
the existence question for these graphs.
Keywords: Locally s-arc-transitive graph, quasiprimitive group, product action.
Math. Subj. Class. (2020): 20B25, 05C25, 05E18
*This paper formed part of the Australian Research Council’s Discovery Project DP120100446 of the first au-
thor. The authors would also like to thank Ákos Seress, who made this collaboration possible in 2012 by allowing
the second author to visit Australia, and Luke Morgan, for providing a proof of Lemma 3.2 and encouraging the
completion of the project. Finally, the authors wish to thank the anonymous referees for their detailed comments
and suggestions that greatly improved the final version of this paper.
†Corresponding author.
E-mail addresses: michael.giudici@uwa.edu.au (Michael Giudici), easwartz@wm.edu (Eric Swartz)
cb This work is licensed under https://creativecommons.org/licenses/by/4.0/
ISSN 1855-3966 (tiskana izd.), ISSN 1855-3974 (elektronska izd.)
ARS MATHEMATICA CONTEMPORANEA 23 (2023) #P2.10
https://doi.org/10.26493/1855-3974.2857.07b
(Dostopno tudi na http://amc-journal.eu)
Lokalno s-ločno tranzitivni grafi, ki nastanejo iz
produktnega delovanja*
Michael Giudici
Department of Mathematics and Statistics, The University of Western Australia,
Perth WA 6009, Australia
Eric Swartz †
Department of Mathematics, William & Mary,
P.O. Box 8795, Williamsburg, VA 23187-8795, USA
Prejeto 30. marca 2022, sprejeto 14. septembra 2022, objavljeno na spletu 13. decembra 2022
Povzetek
Raziskujemo lokalno s-ločno tranzitivne grafe, ki jih dobimo iz kvaziprimitivnega pro-
duktnega delovanja (PA). Dokažemo, da za poljuben lokalno (G, 2)-ločno tranzitiven graf,
pri katerem grupa G deluje kvaziprimitivno s tipom PA na obeh G-orbitah točk, grupa G
ne deluje primitivno na nobeni od orbit. Še več, konstruiramo prve primere lokalno s-ločno
tranzitivnih grafov tipa PA, ki niso standardni dvojni krovi s-ločno tranzitivnih grafov tipa
PA, s čimer je odgovorjeno na eksistenčno vprašanje v zvezi s temi grafi.
Ključne besede: Lokalno s-ločno tranzitiven graf, kvaziprimitivna grupa, produktno delovanje.
Math. Subj. Class. (2020): 20B25, 05C25, 05E18
*Ta članek je tvoril del raziskave Australian Research Council’s Discovery Project DP120100446 prvega av-
torja. Avtorja bi se rada zahvalila tudi Ákosu Seressu, ki je omogočil to sodelovanje v letu 2012, ko je drugi avtor
lahko obiskal Avstralijo, in Luku Morganu, ki je priskrbel dokaz Leme 3.2 in spodbujal k dokončanju projekta.
Avtorja se želita zahvaliti tudi neznanim recenzentom za njihove podrobne pripombe in predloge, ki so znatno
pripomogli k boljši končni različici tega članka.
†Kontaktni avtor.
E-poštna naslova: michael.giudici@uwa.edu.au (Michael Giudici), easwartz@wm.edu (Eric Swartz)
cb To delo je objavljeno pod licenco https://creativecommons.org/licenses/by/4.0/