{"?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-FZ1D4JM5/6ff49158-b7a0-4dfc-9b6e-13202ec02698/PDF","dcterms:extent":"418 KB"},{"@rdf:about":"http://www.dlib.si/stream/URN:NBN:SI:doc-FZ1D4JM5/13113e75-fd02-47ad-953e-a0b45e6aa9d5/TEXT","dcterms:extent":"51 KB"},{"@rdf:about":"http://www.dlib.si/stream/URN:NBN:SI:doc-FZ1D4JM5/164e5456-4bdc-4442-9c98-82d2e9ee1506/PDF","dcterms:extent":"155 KB"},{"@rdf:about":"http://www.dlib.si/stream/URN:NBN:SI:doc-FZ1D4JM5/644d7e50-f7c7-44e7-a3c6-10bce9dd0f68/TEXT","dcterms:extent":"5 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-FZ1D4JM5","dcterms:isPartOf":[{"@rdf:resource":"https://www.dlib.si/details/URN:NBN:SI:spr-UP1WMFAR"},{"@xml:lang":"sl","#text":"Ars mathematica contemporanea"}],"dcterms:issued":"2023","dc:creator":["Burgess, Andrea","Cavenagh, Nicholas J.","Pike, David A. Pike"],"dc:format":[{"@xml:lang":"sl","#text":"številka:2"},{"@xml:lang":"sl","#text":"letnik:23"},{"@xml:lang":"sl","#text":"P2.05 (20 str.)"}],"dc:identifier":["DOI:10.26493/1855-3974.2692.86d","COBISSID_HOST:151481603","ISSN:1855-3966","URN:URN:NBN:SI:doc-FZ1D4JM5"],"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":"ciklični sistemi ciklov"},{"@xml:lang":"en","#text":"completely-reducible"},{"@xml:lang":"en","#text":"cyclic cycle systems"},{"@xml:lang":"en","#text":"Heffter arrays"},{"@xml:lang":"sl","#text":"Heffterjeva polja"},{"@xml:lang":"en","#text":"orthogonal cycle decompositions"},{"@xml:lang":"sl","#text":"popolnoma reducibilen"},{"@xml:lang":"sl","#text":"pravokotne ciklične dekompozicije"},{"@xml:lang":"sl","#text":"superenostaven"},{"@xml:lang":"en","#text":"super-simple"}],"dcterms:temporal":{"@rdf:resource":"2008-2025"},"dc:title":{"@xml:lang":"sl","#text":"Mutually orthogonal cycle systems|"},"dc:description":[{"@xml:lang":"sl","#text":"An ?$\\ell$?-cycle system ?$\\mathcal{F}$? of a graph ?$\\Gamma$? is a set of ?$\\ell$?-cycles which partition the edge set of ?$\\Gamma$?. Two such cycle systems ?$\\mathcal{F}$? and ?$\\mathcal{F}'$? are said to be orthogonal if no two distinct cycles from ?$\\mathcal{F} \\cup \\mathcal{F}'$? share more than one edge. Orthogonal cycle systems naturally arise from face 2-colourable polyehdra and in higher genus from Heffter arrays with certain orderings. A set of pairwise orthogonal ?$\\ell$?-cycle systems of ?$\\Gamma$? is said to be a set of mutually orthogonal cycle systems of ?$\\Gamma$?. Let ?$\\mu(\\ell, n)$? (respectively, ?$\\mu'(\\ell, n)$?) be the maximum integer ?$\\mu$? such that there exists a set of ?$\\mu$? mutually orthogonal (cyclic) v$\\ell$?-cycle systems of the complete graph ?$K_n$?. We show that if ?$\\ell \\ge 4$? is even and v$n \\equiv 1 \\pmod 2\\ell$?, then ?$\\mu'(\\ell,n)$?, and hence ?$\\mu(\\ell,n)$?, is bounded below by a constant multiple of ?$n/\\ell^2$?. In contrast, we obtain the following upper bounds: ?$\\mu(\\ell,n) \\le n-2$?; ?$\\mu(\\ell,n) \\le (n-2)(n-3)/(2(\\ell-3))$? when ?$\\ell \\ge 4$?; ?$\\mu(\\ell, n) \\le 1$? when ?$\\ell > n/\\sqrt{2}$?; and ?$\\mu'(\\ell,n) \\le n-3$? when ?$n \\ge 4$?. We also obtain computational results for small values of ?$n$? and ?$\\ell$?"},{"@xml:lang":"sl","#text":"?$\\ell$?-ciklični sistem ?$\\mathcal{F}$? grafa ?$\\Gamma$? je množica ?$\\ell$?-ciklov, ki razdelijo množico povezav grafa ?$\\Gamma$?. Dva takšna ciklična sistema ?$\\mathcal{F}$? in ?$\\mathcal{F}'$? sta medsebojno pravokotna, če si nobena dva različna cikla iz ?$\\mathcal{F} \\cup \\mathcal{F}'$? ne delita več kot ene povezave. Pravokotni sistemi ciklov nastanejo naravno iz poliedrov z 2-barvnim barvanjem lic, pri ploskvah višjega rodu pa iz Heffterjevih polj, ki zadoščajo določenim pogojem. Množica paroma pravokotnih ?$\\ell$?-cikličnih sistemov grafa ?$\\Gamma$? je množica medsebojno pravokotnih cikličnih sistemov grafa ?$\\Gamma$?. Naj bo ?$\\mu(\\ell, n)$? (oziroma, ?$\\mu'(\\ell, n)$?) maksimalno celo število ?$\\mu$?, pri katerem obstaja množica ?$\\mu$? medsebojno pravokotnih (cikličnih) sistemov ?$\\ell$?-ciklov polnega grafa ?$K_n$?. Dokažemo: če je ?$\\ell \\ge 4$? sod in ?$n \\equiv 1 \\pmod 2\\ell$?, potem je ?$\\mu'(\\ell,n)$?, in torej ?$\\mu(\\ell,n)$?, omejen navzdol s konstantnim večkratnikom števila ?$n/\\ell^2$?. Dobimo tudi naslednje zgornje meje: ?$\\mu(\\ell,n) \\le n-2$?; ?$\\mu(\\ell,n) \\le (n-2)(n-3)/(2(\\ell-3))$?, če je ?$\\ell \\ge 4$?; ?$\\mu(\\ell, n) \\le 1$?, če je ?$\\ell > n/\\sqrt{2}$?; in ?$\\mu'(\\ell,n) \\le n-3$?, če je ?$n \\ge 4$?. Predstavimo tudi računske rezultate za majhne vrednosti ?$n$? in ?$\\ell$?"}],"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-FZ1D4JM5","edm:aggregatedCHO":{"@rdf:resource":"URN:NBN:SI:doc-FZ1D4JM5"},"edm:isShownBy":{"@rdf:resource":"http://www.dlib.si/stream/URN:NBN:SI:doc-FZ1D4JM5/6ff49158-b7a0-4dfc-9b6e-13202ec02698/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-FZ1D4JM5/maxi/edm"},"edm:isShownAt":{"@rdf:resource":"http://www.dlib.si/details/URN:NBN:SI:doc-FZ1D4JM5"}}}}