{"?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-OJAABSC1/019f9aba-df4c-459b-80be-6df94627240a/PDF","dcterms:extent":"454 KB"},{"@rdf:about":"http://www.dlib.si/stream/URN:NBN:SI:doc-OJAABSC1/288a6381-5653-4fcf-a02a-7aa26601f6db/TEXT","dcterms:extent":"71 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-OJAABSC1","dcterms:isPartOf":[{"@rdf:resource":"https://www.dlib.si/details/URN:NBN:SI:spr-UP1WMFAR"},{"@xml:lang":"sl","#text":"Ars mathematica contemporanea"}],"dcterms:issued":"2022","dc:creator":["Ollis, M. A.","Pasotti, Anita","Pellegrini, Marco Antonio","Schmitt, John R."],"dc:format":[{"@xml:lang":"sl","#text":"letnik:22"},{"@xml:lang":"sl","#text":"številka:4"},{"@xml:lang":"sl","#text":"str. 567-594"}],"dc:identifier":["DOI:10.26493/1855-3974.2659.be1","COBISSID_HOST:141370627","ISSN:1855-3966","URN:URN:NBN:SI:doc-OJAABSC1"],"dc:language":"en","dc:publisher":{"@xml:lang":"sl","#text":"Univerza na Primorskem, Fakulteta za matematiko, naravoslovje in informacijske tehnologije"},"dc:subject":[{"@xml:lang":"en","#text":"complete graph"},{"@xml:lang":"sl","#text":"dolžina povezave"},{"@xml:lang":"en","#text":"edge-length"},{"@xml:lang":"en","#text":"growable realization"},{"@xml:lang":"en","#text":"Hamiltonian path"},{"@xml:lang":"sl","#text":"Hamiltonova pot"},{"@xml:lang":"sl","#text":"polni graf"},{"@xml:lang":"sl","#text":"rastoča realizacija"}],"dcterms:temporal":{"@rdf:resource":"2008-2025"},"dc:title":{"@xml:lang":"sl","#text":"Growable realizations: a powerful approach to the Buratti-Horak-Rosa Conjecture|"},"dc:description":[{"@xml:lang":"sl","#text":"Label the vertices of the complete graph ?$K_v$? with the integers ?$\\{0, 1, \\dots, v-1\\}$? and define the length of the edge between the vertices ?$x$? and ?$y$? to be ?$\\min (|x-y|,v-|x-y|)$?. Let ?$L$? be a multiset of size ?$v-1$? with underlying set contained in ?$\\{1, \\dots, \\lfloor v/2 \\rfloor \\}$?. The Buratti-Horak-Rosa Conjecture is that there is a Hamiltonian path in ?$K_v$? whose edge lengths are exactly ?$L$? if and only if for any divisor ?$d$? of ?$v$? the number of multiples of ?$d$? appearing in ?$L$? is at most ?$v-d$?. We introduce \"growable realizations\", which enable us to prove many new instances of the conjecture and to reprove known results in a simpler way. As examples of the new method, we give a complete solution when the underlying set is contained in ?$\\{1, 4, 5\\}$? or in ?$\\{1, 2, 3, 4\\}$? and a partial result when the underlying set has the form ?$\\{1, x, 2x\\}$?. We believe that for any set ?$U$? of positive integers there is a finite set of growable realizations that implies the truth of the Buratti-Horak-Rosa Conjecture for all but finitely many multisets with underlying set ?$U$?"},{"@xml:lang":"sl","#text":"Označimo vozlišča polnega grafa ?$K_v$? s celimi števili ?$\\{0, 1, \\dots, v-1\\}$? in definirajmo dolžino povezave med vozliščema ?$x$? in ?$y$? kot ?$\\min (|x-y|,v-|x-y|)$?. Naj bo ?$L$? multimnožica velikosti ?$v-1$? z osnovno množico, vsebovano v množici ?$\\{1, \\dots, \\lfloor v/2 \\rfloor \\}$?. Buratti-Horak-Roseova domneva pravi, da obstaja Hamiltonova pot v polnem grafu ?$K_v$?, za katero velja, da dolžine njenih povezav tvorijo natanko multimnožico ?$L$?, če in samo če je za poljuben delitelj ?$d$? števila ?$v$? število večkratnikov števila ?$d$?, ki nastopajo v množici ?$L$?, največ ?$v-d$?. Vpeljemo t.i. rastoče realizacije, ki nam omogočajo dokazati mnoge nove primere te domneve in na novo dokazati znane rezultate enostavneje. Predstavimo dva primera uporabe te nove metode: popolno rešitev, če je osnovna množica vsebovana v ?$\\{1, 4, 5\\}$? ali v ?$\\{1, 2, 3, 4\\}$?, in delni rezultat, če ima osnovna množica obliko ?$\\{1, x, 2x\\}$?. Verjamemo, da za vsako množico ?$U$? pozitivnih celih števil obstaja končna množica rastočih realizacij, ki implicira resničnost Buratti-Horak-Roseove domneve za vse multimnožice z osnovno množico ?$U$?, z izjemo končno mnogo takšnih multimnožic"}],"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-OJAABSC1","edm:aggregatedCHO":{"@rdf:resource":"URN:NBN:SI:doc-OJAABSC1"},"edm:isShownBy":{"@rdf:resource":"http://www.dlib.si/stream/URN:NBN:SI:doc-OJAABSC1/019f9aba-df4c-459b-80be-6df94627240a/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-OJAABSC1/maxi/edm"},"edm:isShownAt":{"@rdf:resource":"http://www.dlib.si/details/URN:NBN:SI:doc-OJAABSC1"}}}}