ISSN 2590-9770
The Art of Discrete and Applied Mathematics 3 (2020) #P1.07
https://doi.org/10.26493/2590-9770.1284.3ad
(Also available at http://adam-journal.eu)
Reflexible complete regular dessins and
antibalanced skew morphisms of cyclic groups
Kan Hu*
Department of Mathematics, Zhejiang Ocean University,
Zhoushan, Zhejiang 316022, P.R. China
Young Soo Kwon†
Department of Mathematics, Yeungnam University,
Kyeongsan, 712-749, Republic of Korea
Received 27 December 2018, accepted 19 September 2019, published online 3 August 2020
Abstract
A skew morphism of a finite group A is a bijection ϕ on A fixing the identity element
of A and for which there exists an integer-valued function π on A such that ϕ(ab) =
ϕ(a)ϕπ(a)(b), for all a, b ∈ A. In addition, if ϕ−1(a) = ϕ(a−1)−1, for all a ∈ A,
then ϕ is called antibalanced. In this paper we develop a general theory of antibalanced
skew morphisms and establish a one-to-one correspondence between reciprocal pairs of
antibalanced skew morphisms of the cyclic additive groups and isomorphism classes of
reflexible regular dessins with complete bipartite underlying graphs. As an application,
reflexible complete regular dessins are classified.
Keywords: Graph embedding, antibalanced skew morphism, reciprocal pair.
Math. Subj. Class.: 20B25, 05C10, 14H57
*Author was supported by Natural Science Foundation of Zhejiang Province (LY16A010010, LQ17A010003)
and National Natural Science Foundation of China (11801507).
†Author is supported by the Basic Science Research Program through the National Research Foundation of
Korea (NRF) funded by the Ministry of Education (2018R1D1A1B05048450).
E-mail addresses: hukan@zjou.edu.cn (Kan Hu), ysookwon@ynu.ac.kr (Young Soo Kwon)
cb This work is licensed under https://creativecommons.org/licenses/by/4.0/
ISSN 2590-9770
The Art of Discrete and Applied Mathematics 3 (2020) #P1.07
https://doi.org/10.26493/2590-9770.1284.3ad
(Dostopno tudi na http://adam-journal.eu)
Refleksibilne polne regularne risbe in
neuravnoteženi poševni morfizmi cikličnih grup
Kan Hu*
Department of Mathematics, Zhejiang Ocean University,
Zhoushan, Zhejiang 316022, P.R. China
Young Soo Kwon†
Department of Mathematics, Yeungnam University,
Kyeongsan, 712-749, Republic of Korea
Prejeto 27. decembra 2018, sprejeto 19. septembra 2019, objavljeno na spletu 3. avgusta 2020
Povzetek
Poševni morfizem končne grupe A je bijekcija ϕ na A, ki fiksira identiteto grupe A
in za katero obstaja funkcija s celoštevilskimi vrednostmi π na A tako da je ϕ(ab) =
ϕ(a)ϕπ(a)(b), za vse a, b ∈ A. Če poleg tega velja ϕ−1(a) = ϕ(a−1)−1, za vse a ∈
A, potem se ϕ imenuje neuravnotežen. V tem članku razvijemo splošno teorijo neu-
ravnoteženih poševnih morfizmov in vpeljemo bijektivno korespondenco med reciproč-
nimi pari neuravnoteženih poševnih morfizmov cikličnih aditivnih grup in izomorfnostnimi
razredi refleksibilnih regularnih risb s polnimi dvodelnimi osnovnimi grafi. Kot primer ap-
likacije te teorije klasificiramo refleksibilne polne regularne risbe.
Ključne besede: Vložitev grafa, neuravnotežen poševni morfizem, recipročni par.
Math. Subj. Class.: 20B25, 05C10, 14H57
*Avtor je bil podprt s strani Natural Science Foundation of Zhejiang Province (LY16A010010, LQ17A010003)
in National Natural Science Foundation of China (11801507).
†Avtor je podprt s strani Basic Science Research Program preko National Research Foundation of Korea
(NRF), financiranega od Ministry of Education (2018R1D1A1B05048450).
E-poštna naslova: hukan@zjou.edu.cn (Kan Hu), ysookwon@ynu.ac.kr (Young Soo Kwon)
cb To delo je objavljeno pod licenco https://creativecommons.org/licenses/by/4.0/