{"?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-1GSMAYPU/9c88af4a-6684-4607-bf21-1ce32473b563/PDF","dcterms:extent":"509 KB"},{"@rdf:about":"http://www.dlib.si/stream/URN:NBN:SI:doc-1GSMAYPU/77ca960e-e102-459e-b739-64dd8144209f/TEXT","dcterms:extent":"71 KB"},{"@rdf:about":"http://www.dlib.si/stream/URN:NBN:SI:doc-1GSMAYPU/ef2782ad-f901-489a-a2bc-1a9180554ea8/PDF","dcterms:extent":"142 KB"},{"@rdf:about":"http://www.dlib.si/stream/URN:NBN:SI:doc-1GSMAYPU/7eeec449-9b56-4b2c-a9a3-028319981196/TEXT","dcterms:extent":"4 KB"}],"edm:TimeSpan":{"@rdf:about":"2008-2026","edm:begin":{"@xml:lang":"en","#text":"2008"},"edm:end":{"@xml:lang":"en","#text":"2026"}},"edm:ProvidedCHO":{"@rdf:about":"URN:NBN:SI:doc-1GSMAYPU","dcterms:isPartOf":[{"@rdf:resource":"https://www.dlib.si/details/URN:NBN:SI:spr-UP1WMFAR"},{"@xml:lang":"sl","#text":"Ars mathematica contemporanea"}],"dcterms:issued":"2026","dc:creator":["Agirseven, Onur","Ollis, M. A."],"dc:format":[{"@xml:lang":"sl","#text":"letnik:26"},{"@xml:lang":"sl","#text":"številka:4, article  p4.07"},{"@xml:lang":"sl","#text":"str. 1-27"}],"dc:identifier":["DOI:10.26493/1855-3974.3348.c69","ISSN:1855-3974","COBISSID_HOST:294225923","URN:URN:NBN:SI:doc-1GSMAYPU"],"dc:language":"en","dc:publisher":{"@xml:lang":"sl","#text":"Univerza na Primorskem"},"dc:subject":[{"@xml:lang":"en","#text":"complete graph"},{"@xml:lang":"sl","#text":"dolžina povezave"},{"@xml:lang":"en","#text":"edge-length"},{"@xml:lang":"en","#text":"grid-based graph"},{"@xml:lang":"en","#text":"Hamiltonian path"},{"@xml:lang":"sl","#text":"Hamiltonova pot"},{"@xml:lang":"sl","#text":"mrežasti graf"},{"@xml:lang":"sl","#text":"polni graf"},{"@xml:lang":"sl","#text":"realizacija"},{"@xml:lang":"en","#text":"realization"}],"dcterms:temporal":{"@rdf:resource":"2008-2026"},"dc:title":{"@xml:lang":"sl","#text":"Grid-based graphs, linear realizations, and 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, …, v - 1} and define the length l of the edge between distinct vertices labeled x and y by l(x, y) = min (|y - x|,v - |y - x|). A realization of a multiset L of size v - 1 is a Hamiltonian path through K_v whose edge labels are L. The Buratti-Horak-Rosa (BHR) Conjecture is that there is a realization for a multiset L if and only if for any divisor d of v the number of multiples of d in L is at most v - d. We introduce “grid-based graphs\" as a useful tool for constructing particular types of realizations, called “linear realizations,\"especially when the multiset in question has a support of size 3. This lets us prove many new instances of the BHR Conjecture, including those for multisets of the form {1^a, x^b, y^c} when a ? x + y - ?, where ? is the number of even elements in {x, y}, and those for all multisets of the following forms for sufficiently large v with gcd (v,y) = 1 for all y ? L: •{1^a, 2^b, x^c}, except possibly when a ? {1, 2} and x is odd, •{1^a, x^b, (x + 1)^c}. This establishes that there are infinitely many sets U of size 3 for which there are infinitely many values of v where the BHR Conjecture holds for each multiset with support U. We also show that the BHR Conjecture holds for {1^a, x^b, (x + 1)^c} when x ? {7, 9, 10} and gcd (v, x) = gcd (v, x + 1) = 1"},{"@xml:lang":"sl","#text":"Označimo vozlišča polnega grafa K_v s celimi števili iz množice {0, 1, …, v - 1} in definirajmo dolžino l povezave med različnima vozliščema, označenima z x in y, kot l(x, y) = min(|y - x|, v - |y - x|). Realizacija multimnožice L velikosti v - 1 je Hamiltonova pot v grafu K_v, katere oznake povezav tvorijo množico L. Buratti–Horak–Rosova (BHR) domneva pravi, da realizacija multimnožice L obstaja natanko tedaj, ko za vsak delitelj d števila v število veˇckratnikov števila d v L ni večje od v - d. Vpeljemo “mrežaste grafe” kot uporabno orodje za konstrukcijo posebnih vrst realizacij, imenovanih “linearne realizacije”, zlasti kadar ima obravnavana multimnožica nosilec velikosti 3. S tem dokažemo številne nove primere BHR-domneve, med drugim za multimnožice oblike {1^a, x^b, y^c}, kadar velja a ? x + y - ?, kjer je ? število sodih elementov v množici {x, y}, ter za vse multimnožice naslednjih oblik pri dovolj velikem v, če velja gcd(v, y) = 1 za vsak y ? L: • {1^a, 2^b, x^c}, razen morda kadar je a ? {1, 2} in je x liho število, • {1^a, x^b, (x + 1)^c}. S tem pokažemo, da obstaja neskončno mnogo množic U velikosti 3, za katere obstaja neskončno mnogo vrednosti v, pri katerih BHR-domneva velja za vsako multimnožico z nosilcem U . Poleg tega pokažemo, da BHR-domneva velja za {1^a, x^b, (x + 1)^c}, kadar je x ? {7, 9, 10} in gcd(v, x) = gcd(v, x + 1) = 1"}],"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-1GSMAYPU","edm:aggregatedCHO":{"@rdf:resource":"URN:NBN:SI:doc-1GSMAYPU"},"edm:isShownBy":{"@rdf:resource":"http://www.dlib.si/stream/URN:NBN:SI:doc-1GSMAYPU/9c88af4a-6684-4607-bf21-1ce32473b563/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-1GSMAYPU/maxi/edm"},"edm:isShownAt":{"@rdf:resource":"http://www.dlib.si/details/URN:NBN:SI:doc-1GSMAYPU"}}}}