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Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2017, Volume 23, Number 4, Pages 243–252
DOI: https://doi.org/10.21538/0134-4889-2017-23-4-243-252
(Mi timm1483)
 

On the Oikawa and Arakawa theorems for graphs

A. D. Mednykha, I. A. Mednykha, R. Nedelyabc

a Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences, Novosbirsk, 630090 Russia
b University of West Bohemia, NTIS FAV, Universitni 8, Pilsen, Czech Republic
c Matej Bel University, Tajovskeho 40, Banska Bystrica, Slovakia
References:
Abstract: The present paper is devoted to the further development of the discrete theory of Riemann surfaces, which was started in the papers by M. Baker and S. Norine at the beginning of the century. This theory considers finite graphs as analogs of compact Riemann surfaces and branched coverings of graphs as holomorphic maps. The genus of a graph is defined as the rank of its fundamental group. The main object of investigation in the paper is automorphism groups of a graph acting freely on the set of arcs. These groups are discrete analogs of groups of conformal automorphisms of a Riemann surface. The celebrated Hurwitz theorem (1893) states that the order of the group of conformal automorphisms of a compact Riemann surface of genus $g>1$ does not exceed $84(g-1)$. Later, K. Oikawa and T. Arakawa refined this bound in the case of groups that fix several finite sets of prescribed cardinalities. This paper provides proofs of discrete versions of the mentioned theorems. In addition, a graph-theoretic version of the E. Bujalance and G. Gromadzki result improving the Arakawa theorem is obtained.
Keywords: Riemann surface, Riemann–Hurwitz formula, graph, automorphism group, harmonic map.
Funding agency Grant number
Russian Foundation for Basic Research 15-01-07906
16-31-00138
Ministry of Education, Youth and Sports of the Czech Republic L01506
Czech Science Foundation P202/12/G061
The research of R. Nedela was partially supported by the project L01506 of the Czech Ministry of Education, Youth and Sports and by the project P202/12/G061 of Czech Science Foundation.
Received: 14.06.2017
English version:
Proceedings of the Steklov Institute of Mathematics (Supplementary issues), 2019, Volume 304, Issue 1, Pages S133–S140
DOI: https://doi.org/10.1134/S0081543819020147
Bibliographic databases:
Document Type: Article
UDC: 519.177+517.545
MSC: 05C10, 57M12
Language: Russian
Citation: A. D. Mednykh, I. A. Mednykh, R. Nedelya, “On the Oikawa and Arakawa theorems for graphs”, Trudy Inst. Mat. i Mekh. UrO RAN, 23, no. 4, 2017, 243–252; Proc. Steklov Inst. Math. (Suppl.), 304, suppl. 1 (2019), S133–S140
Citation in format AMSBIB
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\by A.~D.~Mednykh, I.~A.~Mednykh, R.~Nedelya
\paper On the Oikawa and~Arakawa~theorems for graphs
\serial Trudy Inst. Mat. i Mekh. UrO RAN
\yr 2017
\vol 23
\issue 4
\pages 243--252
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\crossref{https://doi.org/10.21538/0134-4889-2017-23-4-243-252}
\elib{https://elibrary.ru/item.asp?id=30713977}
\transl
\jour Proc. Steklov Inst. Math. (Suppl.)
\yr 2019
\vol 304
\issue , suppl. 1
\pages S133--S140
\crossref{https://doi.org/10.1134/S0081543819020147}
\isi{https://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=Publons&SrcAuth=Publons_CEL&DestLinkType=FullRecord&DestApp=WOS_CPL&KeyUT=000453521700022}
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