ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 24 (2024) #P1.06
https://doi.org/10.26493/1855-3974.2503.f17
(Also available at http://amc-journal.eu)
Classification of thin regular map
representations of hypermaps*
Antonio Breda d’Azevedo , Domenico A. Catalano †
Department of Mathematics, University of Aveiro, Aveiro, Portugal
Received 12 December 2020, accepted 13 January 2023, published online 22 August 2023
Abstract
There are two well-known map representations of hypermaps, namely the Walsh and
the Vince map representations, that are dual to each other. They correspond to normal sub-
groups of index two of a free product Γ = (C2×C2)∗C2 which decompose as “elementary”
free product C2 ∗C2 ∗C2. However, Γ has three normal subgroups that decompose as “el-
ementary" free product C2 ∗ C2 ∗ C2, the third of these subgroups giving the less known
Petrie-path map representation. By relaxing the “elementary" free product condition to free
product of rank 3, and under the extra condition “words of smaller length" on the genera-
tors, we prove that the number of map representations of hypermaps increases to 15 (up to
a restrictedly dual), all of which are described in this paper.
Keywords: Map representation, hypermaps, maps, regularity, restricted regularity, orientably regular.
Math. Subj. Class. (2020): 05C10, 05C25, 05C65, 05E18, 20F65
*The authors thank the support given by the Portuguese funds through the CIDMA - Center for Research and
Development in Mathematics and Applications and the Portuguese Foundation for Science and Technology (FCT-
Fundação para a Ciência e a Tecnologia) within project UIDB/04106/2020.
†Corresponding author.
E-mail addresses: breda@ua.pt (Antonio Breda d’Azevedo), domenico@ua.pt (Domenico A. Catalano)
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) #P1.06
https://doi.org/10.26493/1855-3974.2503.f17
(Dostopno tudi na http://amc-journal.eu)
Klasifikacija tankih regularnih predstavitev
hiperzemljevidov z zemljevidi*
Antonio Breda d’Azevedo , Domenico A. Catalano †
Department of Mathematics, University of Aveiro, Aveiro, Portugal
Prejeto 12. decembra 2020, sprejeto 13. januarja 2023, objavljeno na spletu 22. avgusta 2023
Povzetek
Obstajata dve dobro znani predstavitvi hiperzemljevidov z zemljevidi, in sicer Wal-
sheva in Vinceova; ti dve predstavitvi sta druga drugi dualni. Ustrezata dvema podgrupama
edinkama indeksa dve prostega produkta Γ = (C2×C2)∗C2, ki razpade na “elementarni”
prosti produkt C2 ∗ C2 ∗ C2. Vendar pa ima Γ tri podgrupe edinke, ki razpadejo na “ele-
mentarni" prosti produkt C2 ∗C2 ∗C2, pri čemer tretja od teh podgrup edink ustreza manj
znani zemljevidni predstavitvi z uporabo Petriejevih poti. Če umaknemo pogoj, da mora
biti prosti produkt “elementaren" oz. oblike C2 ∗ C2 ∗ C2, in zahtevamo le, da ima rang
3, ter ob dodatnem pogoju, da so “besede iz generatorjev krajše", dokažemo, da število
predstavitev hiperzemljevidov z zemljevidi naraste na 15 (do omejenega duala natančno);
in prav vse takšne predstavitve so prikazane v tem članku.
Ključne besede: Predstavitev z zemljevidi, hiperzemljevidi, zemljevidi, regularnost, omejena regu-
larnost, orientabilno regularen.
Math. Subj. Class. (2020): 05C10, 05C25, 05C65, 05E18, 20F65
*Avtorja se zahvaljujeta za podporo Portuguese funds preko CIDMA - Center for Research and Development
in Mathematics and Applications ter za podporo Portuguese Foundation for Science and Technology (FCT- Fun-
dação para a Ciência e a Tecnologia) v okviru projakta UIDB/04106/2020.
†Kontaktni avtor.
E-poštna naslova: breda@ua.pt (Antonio Breda d’Azevedo), domenico@ua.pt (Domenico A. Catalano)
cb To delo je objavljeno pod licenco https://creativecommons.org/licenses/by/4.0/