ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 24 (2024) #P1.01
https://doi.org/10.26493/1855-3974.3071.37e
(Also available at http://amc-journal.eu)
Almost all Cayley maps are mapical regular
representations
Pablo Spiga *
Dipartimento di Matematica e Applicazioni, University of Milano-Bicocca,
Via Cozzi 55, 20125 Milano, Italy
Dario Sterzi
Scuola Galileiana di Studi Superiori, Via Venezia 20, 35131, Padova, Italy and
Dipartimento di Matematica “Tullio Levi-Civita”,
University of Padova, Via Trieste 53, 35121 Padova, Italy
Received 13 February 2023, accepted 5 May 2023, published online 6 July 2023
Abstract
Cayley maps are combinatorial structures built upon Cayley graphs on a group. As
such the original group embeds in their group of automorphisms, and one can ask in which
situation the two coincide (one then calls the Cayley map a mapical regular representation
or MRR) and with what probability. The first question was answered by Jajcay. In this
paper we tackle the probabilistic version, and prove that as groups get larger the proportion
of MRRs among all Cayley Maps approaches 1.
Keywords: Regular representation, Cayley map, automorphism group, asymptotic enumeration, graph-
ical regular representation, GRR.
Math. Subj. Class. (2020): 05C25, 05C30, 20B25, 20B15
*Corresponding author.
E-mail addresses: pablo.spiga@unimib.it (Pablo Spiga), dario.sterzi@studenti.unipd.it (Dario Sterzi)
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) #P1.01
https://doi.org/10.26493/1855-3974.3071.37e
(Dostopno tudi na http://amc-journal.eu)
Skoraj vsi Cayleyjevi zemljevidi so zemljevidne
regularne predstavitve
Pablo Spiga *
Dipartimento di Matematica e Applicazioni, University of Milano-Bicocca,
Via Cozzi 55, 20125 Milano, Italy
Dario Sterzi
Scuola Galileiana di Studi Superiori, Via Venezia 20, 35131, Padova, Italy in
Dipartimento di Matematica “Tullio Levi-Civita”,
University of Padova, Via Trieste 53, 35121 Padova, Italy
Prejeto 13. februarja 2023, sprejeto 5. maja 2023, objavljeno na spletu 6. julija 2023
Povzetek
Cayleyjevi zemljevidi so kombinatorične strukture, zgrajene na podlagi Cayleyjevih
grafov neke grupe. Ker je izvorna grupa vložljiva v njihovo grupo avtomorfizmov, se lahko
vprašamo, kdaj ti dve grupi sovpadata (Cayleyjev zemljevid tedaj imenujemo zemljevidna
regularna predstavitev ali ZPR), in s kakšno verjetnostjo se to zgodi. Na prvo vprašanje
je odgovoril Jajcay. V tem članku obravnavamo verjetnostno različico in dokažemo, da se
s povečevanjem grup delež zemljevidnih regularnih predstavitev med vsemi Cayleyjevimi
zemljevidi približuje 1.
Ključne besede: Regularna predstavitev, Cayleyjev zemljevid, grupa avtomorfizmov, asimptotska enu-
meracija, grafovska regularna predstavitev, GRP.
Math. Subj. Class. (2020): 05C25, 05C30, 20B25, 20B15
*Kontaktni avtor.
E-poštna naslova: pablo.spiga@unimib.it (Pablo Spiga), dario.sterzi@studenti.unipd.it (Dario Sterzi)
cb To delo je objavljeno pod licenco https://creativecommons.org/licenses/by/4.0/