{"?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-E8D7Z5KO/7c88e4f8-0530-4cc8-ac5f-59788fddc6ff/PDF","dcterms:extent":"367 KB"},{"@rdf:about":"http://www.dlib.si/stream/URN:NBN:SI:doc-E8D7Z5KO/f714559d-1668-4f6f-9eb7-1d5779891c2b/TEXT","dcterms:extent":"37 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-E8D7Z5KO","dcterms:isPartOf":[{"@rdf:resource":"https://www.dlib.si/details/URN:NBN:SI:spr-UP1WMFAR"},{"@xml:lang":"sl","#text":"Ars mathematica contemporanea"}],"dcterms:issued":"2019","dc:creator":"Przybyło, Jakub","dc:format":[{"@xml:lang":"sl","#text":"številka:1"},{"@xml:lang":"sl","#text":"letnik:17"},{"@xml:lang":"sl","#text":"str. 37-49"}],"dc:identifier":["ISSN:1855-3966","COBISSID_HOST:18911833","URN:URN:NBN:SI:doc-E8D7Z5KO"],"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":"adjacent strong chromatic index"},{"@xml:lang":"en","#text":"distant set distinguishing index"},{"@xml:lang":"en","#text":"distant sum distinguishing index of a graph"},{"@xml:lang":"en","#text":"neighbour sum distinguishing index"},{"@xml:lang":"sl","#text":"razlikovalni indeks oddaljenih množic"},{"@xml:lang":"sl","#text":"razlikovalni indeks oddaljenih vsot"},{"@xml:lang":"sl","#text":"razlikovalni indeks sosednjih vsot"},{"@xml:lang":"sl","#text":"sosedni krepki kromatični indeks"}],"dcterms:temporal":{"@rdf:resource":"2008-2025"},"dc:title":{"@xml:lang":"sl","#text":"Distant sum distinguishing index of graphs with bounded minimum degree|"},"dc:description":[{"@xml:lang":"sl","#text":"For any graph ?$G=(V,E)$? with maximum degree ?$\\Delta$? and without isolated edges, and a positive integer ?$r$?, by ?$\\chi'_{\\Sigma,r}(G)$? we denote the ?$r$?-distant sum distinguishing index of ?$G$?. This is the least integer ?$k$? for which a proper edge colouring ?$c:E\\to\\{1,2,\\ldots,k\\}$? exists such that ?$\\sum_{e\\ni u}c(e)\\neq \\sum_{e\\ni v}c(e)$? for every pair of distinct vertices ?$u,v$? at distance at most ?$r$? in ?$G$?. It was conjectured that ?$\\chi'_{\\Sigma,r}(G)\\leq (1+o(1))\\Delta^{r-1}$? for every ?$r\\geq 3$?. Thus far it has been in particular proved that ?$\\chi'_{\\Sigma,r}(G)\\leq 6\\Delta^{r-1}$? if ?$r\\geq 4$?. Combining probabilistic and constructive approach, we show that this can be improved to ?$\\chi'_{\\Sigma,r}(G)\\leq (4+o(1))\\Delta^{r-1}$? if the minimum degree of ?$G$? equals at least ?$\\ln^8\\Delta$?"},{"@xml:lang":"sl","#text":"Za poljuben graf ?$G=(V,E)$? z maksimalno stopnjo ?$\\Delta$? in brez izoliranih povezav ter za poljubno pozitivno celo število ?$r$? označimo s ?$\\chi'_{\\Sigma,r}(G)$? razlikovalni indeks ?$r$?-oddaljenih vsot grafa ?$G$?. To je najmanjše celo število ?$k$?, za katerega obstaja takšno pravilno barvanje povezav ?$c:E\\to\\{1,2,\\ldots,k\\}$?, da velja ?$\\sum_{e\\ni u}c(e)\\neq \\sum_{e\\ni v}c(e)$? za vsak par različnih vozlišč ?$u,v$? na razdalji največ ?$r$? v ?$G$?. Postavljena je bila domneva, da velja ?$\\chi'_{\\Sigma,r}(G)\\leq (1+o(1))\\Delta^{r-1}$? za vsak ?$r\\geq 3$?. Doslej je bilo dokazano, da velja ?$\\chi'_{\\Sigma,r}(G)\\leq 6\\Delta^{r-1}$?, če je ?$r\\geq 4$?. S povezovanjem verjetnostnega in konstruktivnega pristopa pokažemo, da lahko to izboljšamo na ?$\\chi'_{\\Sigma,r}(G)\\leq (4+o(1))\\Delta^{r-1}$?, če je minimalna stopnja grafa ?$G$? enaka najmanj ?$\\ln^8\\Delta$?"}],"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-E8D7Z5KO","edm:aggregatedCHO":{"@rdf:resource":"URN:NBN:SI:doc-E8D7Z5KO"},"edm:isShownBy":{"@rdf:resource":"http://www.dlib.si/stream/URN:NBN:SI:doc-E8D7Z5KO/7c88e4f8-0530-4cc8-ac5f-59788fddc6ff/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-E8D7Z5KO/maxi/edm"},"edm:isShownAt":{"@rdf:resource":"http://www.dlib.si/details/URN:NBN:SI:doc-E8D7Z5KO"}}}}