{"?xml":{"@version":"1.0"},"edm:RDF":{"@xmlns:dc":"http://purl.org/dc/elements/1.1/","@xmlns:edm":"http://www.europeana.eu/schemas/edm/","@xmlns:wgs84_pos":"http://www.w3.org/2003/01/geo/wgs84_pos","@xmlns:foaf":"http://xmlns.com/foaf/0.1/","@xmlns:rdaGr2":"http://rdvocab.info/ElementsGr2","@xmlns:oai":"http://www.openarchives.org/OAI/2.0/","@xmlns:owl":"http://www.w3.org/2002/07/owl#","@xmlns:rdf":"http://www.w3.org/1999/02/22-rdf-syntax-ns#","@xmlns:ore":"http://www.openarchives.org/ore/terms/","@xmlns:skos":"http://www.w3.org/2004/02/skos/core#","@xmlns:dcterms":"http://purl.org/dc/terms/","edm:WebResource":[{"@rdf:about":"http://www.dlib.si/stream/URN:NBN:SI:doc-BV48VQGA/ae539799-4f7a-4fc3-b2dc-13674972486e/PDF","dcterms:extent":"326 KB"},{"@rdf:about":"http://www.dlib.si/stream/URN:NBN:SI:doc-BV48VQGA/cdf3c418-b8bd-4bca-bef0-251aa20892be/TEXT","dcterms:extent":"55 KB"}],"edm:TimeSpan":{"@rdf:about":"2008-2025","edm:begin":{"@xml:lang":"en","#text":"2008"},"edm:end":{"@xml:lang":"en","#text":"2025"}},"edm:ProvidedCHO":{"@rdf:about":"URN:NBN:SI:doc-BV48VQGA","dcterms:isPartOf":[{"@rdf:resource":"https://www.dlib.si/details/URN:NBN:SI:spr-UP1WMFAR"},{"@xml:lang":"sl","#text":"Ars mathematica contemporanea"}],"dcterms:issued":"2020","dc:creator":["Feng, Yan-Quan","Hu, Kan","Nedela, Roman","Škoviera, Martin","Wang, Naer"],"dc:format":[{"@xml:lang":"sl","#text":"letnik:18"},{"@xml:lang":"sl","#text":"številka:2"},{"@xml:lang":"sl","#text":"str. 289-307"}],"dc:identifier":["ISSN:1855-3966","COBISSID_HOST:41168899","URN:URN:NBN:SI:doc-BV48VQGA"],"dc:language":"en","dc:publisher":{"@xml:lang":"sl","#text":"Univerza na Primorskem, Fakulteta za matematiko, naravoslovje in informacijske tehnologije"},"dc:subject":[{"@xml:lang":"sl","#text":"biciklična grupa"},{"@xml:lang":"en","#text":"bicyclic group"},{"@xml:lang":"en","#text":"graph embedding"},{"@xml:lang":"sl","#text":"poševni morfizem"},{"@xml:lang":"sl","#text":"pravilna risba"},{"@xml:lang":"en","#text":"regular dessin"},{"@xml:lang":"en","#text":"skew-morphism"},{"@xml:lang":"sl","#text":"vložitev grafa"}],"dcterms:temporal":{"@rdf:resource":"2008-2025"},"dc:title":{"@xml:lang":"sl","#text":"Complete regular dessins and skew-morphisms of cyclic groups|"},"dc:description":[{"@xml:lang":"sl","#text":"A dessin is a 2-cell embedding of a connected ?$2$?-coloured bipartite graph into an orientable closed surface. A dessin is regular if its group of orientation- and colour-preserving automorphisms acts regularly on the edges. In this paper we study regular dessins whose underlying graph is a complete bipartite graph $K_{m,n}$, called $(m,n)$-complete regular dessins. The purpose is to establish a rather surprising correspondence between ?$(m,n)$?-complete regular dessins and pairs of skew-morphisms of cyclic groups. A skew-morphism of a finite group ?$A$? is a bijection ?$\\varphi\\colon A\\to A$? that satisfies the identity ?$\\varphi(xy)=\\varphi(x)\\varphi^{\\pi(x)}(y)$? for some function ?$\\pi\\colon A\\to\\mathbb{Z}$? and fixes the neutral element of ?$A$?. We show that every ?$(m,n)$?-complete regular dessin ?$\\mathcal{D}$? determines a pair of reciprocal skew-morphisms of the cyclic groups ?$\\mathbb{Z}_n$? and ?$\\mathbb{Z}_m$?. Conversely, ?$\\mathcal{D}$? can be reconstructed from such a reciprocal pair. As a consequence, we prove that complete regular dessins, exact bicyclic groups with a distinguished pair of generators, and pairs of reciprocal skew-morphisms of cyclic groups are all in one-to-one correspondence. Finally, we apply the main result to determining all pairs of integers ?$m$? and ?$n$? for which there exists, up to interchange of colours, exactly one ?$(m,n)$?-complete regular dessin. We show that the latter occurs precisely when every group expressible as a product of cyclic groups of order ?$m$? and ?$n$? is abelian, which eventually comes down to the condition ?$\\gcd(m,\\phi(n))=\\gcd(\\phi(m),n)=1$?, where ?$\\phi$? is Euler's totient function"},{"@xml:lang":"sl","#text":"Risba je 2-celična vložitev povezanega 2-barvnega dvodelnega grafa na orientabilno sklenjeno ploskev. Risba je pravilna, če njena grupa avtomorfizmov, ki ohranjajo orientacijo in barve, deluje pravilno na povezavah. V tem članku preučujemo pravilne risbe, katerih osnovni graf je polni dvodelni graf ?$K_{m,n}$?, imenovane ?$(m,n)$?-polne pravilne risbe. Na ta način vzpostavimo precej presenetljivo korespondenco med ?$(m,n)$?-polnimi pravilnimi risbami in pari poševnih morfizmov cikličnih grup. Poševni morfizem končne grupe ?$A$? je bijekcija ?$\\varphi\\colon A\\to A$?, ki zadošča identiteti ?$\\varphi(xy)=\\varphi(x)\\varphi^{\\pi(x)}(y)$? za neko funkcijo ?$\\pi\\colon A\\to\\mathbb{Z}$? in fiksira nevtralni element grupe ?$A$?. Dokažemo, da vsaka ?$(m,n)$?-polna pravilna risba ?$\\mathcal{D}$? določa par recipročnih poševnih morfizmov cikličnih grup ?$\\mathbb{Z}_n$? in ?$\\mathbb{Z}_m$?. Velja tudi obratno, ?$\\mathcal{D}$? lahko rekonstruiramo iz takšnega recipročnega para. Na podlagi tega dokažemo, da so polne pravilne risbe, eksaktne biciklične grupe z izbranim parom generatorjev, ter pari recipročnih poševnih morfizmov cikličnih grup vsi v povratno enolični korespondenci. Nazadnje pa uporabimo naš glavni rezultat še za določitev vseh parov celih števil ?$m$? in ?$n$?, za katere obstaja, do zamenjave barv natančno, samo en izomorfnostni razred $(m,n)$-polnih regularnih risb. Dokažemo, da se to zgodi natanko takrat, ko je vsaka grupa, izrazljiva kot produkt cikličnih grup reda ?$m$? in ?$n$?, abelska, kar se naposled prevede na pogoj ?$\\gcd(m,\\phi(n))=\\gcd(\\phi(m),n)=1$?, kjer je ?$\\phi$? Eulerjeva funkcija"}],"edm:type":"TEXT","dc:type":[{"@xml:lang":"sl","#text":"znanstveno časopisje"},{"@xml:lang":"en","#text":"journals"},{"@rdf:resource":"http://www.wikidata.org/entity/Q361785"}]},"ore:Aggregation":{"@rdf:about":"http://www.dlib.si/?URN=URN:NBN:SI:doc-BV48VQGA","edm:aggregatedCHO":{"@rdf:resource":"URN:NBN:SI:doc-BV48VQGA"},"edm:isShownBy":{"@rdf:resource":"http://www.dlib.si/stream/URN:NBN:SI:doc-BV48VQGA/ae539799-4f7a-4fc3-b2dc-13674972486e/PDF"},"edm:rights":{"@rdf:resource":"http://creativecommons.org/licenses/by/4.0/"},"edm:provider":"Slovenian National E-content Aggregator","edm:intermediateProvider":{"@xml:lang":"en","#text":"National and University Library of Slovenia"},"edm:dataProvider":{"@xml:lang":"sl","#text":"Univerza na Primorskem, Fakulteta za naravoslovje, matematiko in informacijske tehnologije"},"edm:object":{"@rdf:resource":"http://www.dlib.si/streamdb/URN:NBN:SI:doc-BV48VQGA/maxi/edm"},"edm:isShownAt":{"@rdf:resource":"http://www.dlib.si/details/URN:NBN:SI:doc-BV48VQGA"}}}}