ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 24 (2024) #P3.01
https://doi.org/10.26493/1855-3974.2408.42e
(Also available at http://amc-journal.eu)
Generalized Cayley maps and their Petrie duals*
Robert Jajcay †
Comenius University, Bratislava, Slovakia
Jozef Širáň ‡
Slovak University of Technology, Bratislava, Slovakia
Yan Wang §
Yan Tai University, Yan Tai, China
Received 12 August 2020, accepted 22 October 2023, published online 6 June 2024
Abstract
Cayley maps are embeddings of Cayley graphs in orientable surfaces which possess a
group of orientation preserving automorphisms acting regularly on the vertices. We gen-
eralize the concept of a Cayley map by considering embeddings of Cayley graphs in both
orientable and non-orientable surfaces and by requiring a group of automorphisms acting
regularly on vertices that does not have to consist entirely of orientation preserving auto-
morphisms. This leads to new families of maps in both the orientable and non-orientable
cases. Since the Petrie dual operator preserves the property of being a generalized Cayley
map, throughout the paper we consider the action of this operator on our maps.
Keywords: Cayley map, regular embedding, automorphism group of a map.
Math. Subj. Class. (2020): 05C10, 05C25, 05E18
*We wish to thank our referees for their repeated careful readings of our article and their helpful and knowl-
edgeable suggestions that helped considerably in improving our article and making it more focused.
†Corresponding author. Supported by VEGA Research Grant 1/0437/23 and APVV Research Grant 19-0308.
‡Supported by APVV Research Grants 19-0308 and 22-0005, and VEGA Research Grants 1/0567/22 and
1/0069/23.
§Supported by NSFS No. ZR2020MA044.
E-mail addresses: robert.jajcay@fmph.uniba.sk (Robert Jajcay), jozef.siran@stuba.sk (Jozef Širáň),
wang−yan@pku.org.cn (Yan Wang)
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 24 (2024) #P3.01
https://doi.org/10.26493/1855-3974.2408.42e
(Dostopno tudi na http://amc-journal.eu)
Posplošeni Cayleyjevi zemljevidi in njihovi
Petriejevi duali*
Robert Jajcay †
Comenius University, Bratislava, Slovakia
Jozef Širáň ‡
Slovak University of Technology, Bratislava, Slovakia
Yan Wang §
Yan Tai University, Yan Tai, China
Prejeto 12. avgusta 2020, sprejeto 22. oktobra 2023, objavljeno na spletu 6. junija 2024
Povzetek
Cayleyjevi zemljevidi so vložitve Cayleyjevih grafov na orientabilne ploskve z grupo
avtomorfizmov, ki ohranjajo orientacijo in delujejo regularno na točkah grafa. Posplošimo
koncept Cayleyjevega zemljevida z obravnavanjem vložitev Cayleyjevih grafov tako na
orientabilne kot na neorientabilne ploskve in z zahtevo, da grupa avtomorfizmov deluje
regularno na točkah, pri čemer ni nujno, da sestoji izključno iz avtomorfizmov, ki ohran-
jajo orientacijo. To vodi k novim družinam zemljevidov tako v orientabilnem kot v ne-
orientabilnem primeru. Ker Petriejev dualni operator ohranja lastnost pripadnosti družini
posplošenih Cayleyjevih zemljevidov, v članku obravnavamo delovanje tega operatorja na
naših zemljevidih.
Ključne besede: Cayleyjev zemljevid, regularna vložitev, grupa avtomorfizmov zemljevida.
Math. Subj. Class. (2020): 05C10, 05C25, 05E18
*Želimo se zahvaliti našim recenzentom za njihovo večkratno skrbno branje našega članka in njihove koristne
in strokovne predloge, ki so pomagali znatno izboljšati naš članek in ga naredili bolj osredotočenega.
†Kontaktni avtor. Podprt s strani raziskovalne štipendije VEGA 1/0437/23 in raziskovalne štipendije APVV
19-0308.
‡Podprt s strani raziskovalnih štipendij APVV 19-0308 in 22-0005, in raziskovalnih štipendij VEGA 1/0567/22
in 1/0069/23.
§Podprt s strani NSFS št. ZR2020MA044.
E-poštni naslovi: robert.jajcay@fmph.uniba.sk (Robert Jajcay), jozef.siran@stuba.sk (Jozef Širáň),
wang−yan@pku.org.cn (Yan Wang)
cb To delo je objavljeno pod licenco https://creativecommons.org/licenses/by/4.0/