{"?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-5CVYESBI/dd640e36-b84e-455f-b575-7c5b5e176baa/PDF","dcterms:extent":"343 KB"},{"@rdf:about":"http://www.dlib.si/stream/URN:NBN:SI:doc-5CVYESBI/73bcf9cd-b443-4e61-9d4b-af3987d2c880/TEXT","dcterms:extent":"43 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-5CVYESBI","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":["Hakobyan, Anush","Mkrtchyan, Vahan"],"dc:format":[{"@xml:lang":"sl","#text":"letnik:17"},{"@xml:lang":"sl","#text":"številka:2"},{"@xml:lang":"sl","#text":"str. 431-445"}],"dc:identifier":["ISSN:1855-3966","COBISSID_HOST:18957401","URN:URN:NBN:SI:doc-5CVYESBI"],"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":"?$S_{10}$?-conjecture"},{"@xml:lang":"sl","#text":"?$S_{10}$?-domneva"},{"@xml:lang":"en","#text":"cubic graph"},{"@xml:lang":"sl","#text":"domneva o petersenskem barvanju"},{"@xml:lang":"sl","#text":"kubični graf"},{"@xml:lang":"en","#text":"Petersen coloring conjecture"},{"@xml:lang":"en","#text":"Petersen graph"},{"@xml:lang":"sl","#text":"Petersenov graf"}],"dcterms:temporal":{"@rdf:resource":"2008-2025"},"dc:title":{"@xml:lang":"sl","#text":"Ssub{12} and Psub{12}-colorings of cubic graphs|"},"dc:description":[{"@xml:lang":"sl","#text":"If ?$G$? and ?$H$? are two cubic graphs, then an ?$H$?-coloring of ?$G$? is a proper edge-coloring ?$f$? with the edges of ?$H$?, such that for each vertex ?$x$? of ?$G$?, there is a vertex ?$y$? of ?$H$? with ?$f(\\partial_G(x)) = \\partial_H(y)$?. If ?$G$? admits an ?$H$?-coloring, then we will write ?$H\\prec G$?. The Petersen coloring conjecture of Jaeger (?$P_{10}$?-conjecture) states that for any bridgeless cubic graph ?$G$?, one has: ?$P_{10} \\prec G$?. The ?$S_{10}$?-conjecture states that for any cubic graph ?$G$?, ?$S_{10} \\prec G$?. In this paper, we introduce two new conjectures that are related to these conjectures. The first of them states that any cubic graph with a perfect matching admits an ?$S_{12}$?-coloring. The second one states that any cubic graph ?$G$? whose edge-set can be covered with four perfect matchings, admits a ?$P_{12}$?-coloring. We call these new conjectures ?$S_{12}$?-conjecture and ?$P_{12}$?-conjecture, respectively. Our first results justify the choice of graphs in ?$S_{12}$?-conjecture and ?$P_{12}$?-conjecture. Next, we characterize the edges of ?$P_{12}$? that may be fictive in a ?$P_{12}$?-coloring of a cubic graph ?$G$?. Finally, we relate the new conjectures to the already known conjectures by proving that ?$S_{12}$?-conjecture implies ?$S_{10}$?-conjecture, and ?$P_{12}$?-conjecture and ?$(5, 2)$?-Cycle cover conjecture together imply ?$P_{10}$?-conjecture. Our main tool for proving the latter statement is a new reformulation of ?$(5, 2)$?-Cycle cover conjecture, which states that the edge-set of any claw-free bridgeless cubic graph can be covered with four perfect matchings"},{"@xml:lang":"sl","#text":"Če sta ?$G$? in ?$H$? kubična grafa, potem je ?$H$?-barvanje grafa ?$G$? pravilno povezavno barvanje ?$f$? s povezavami grafa ?$H$?, takšno da za vsako vozlišče ?$x$? grafa ?$G$? obstaja vozlišče ?$y$? grafa ?$H$?, za katero je ?$f(\\partial_G(x)) = \\partial_H(y)$?. Če ?$G$? dopušca ?$H$?-barvanje, potem bomo pisali ?$H\\prec G$?. Jaegerjeva domneva o petersenskem barvanju (?$P_{10}$?-domneva) pravi, da za poljuben brezmostni kubični graf ?$G$? velja ?$P_{10} \\prec G$?. ?$S_{10}$?-domneva pravi, da za poljuben kubični graf ?$G$? velja ?$S_{10} \\prec G$?. V članku vpeljeva dve novi domnevi, ki sta povezani s tema domnevama. Prva od njiju pravi, da poljuben kubični graf s popolnim prirejanjem dopušca ?$S_{12}$?-barvanje. Druga pravi, da poljuben kubični graf ?$G$?, katerega povezavno množico se da pokriti s štirimi popolnimi prirejanji, dopušca ?$P_{12}$?-barvanje. Ti dve novi domnevi imenujeva ?$S_{12}$?-domneva in ?$P_{12}$?-domneva. Najin prvi rezultat opravičuje izbiro grafov v ?$S_{12}$?-domnevi in ?$P_{12}$?-domnevi. Nadalje, karakterizirava povezave v ?$P_{12}$?, ki lahko nastopajo v ?$P_{12}$?-barvanju kubičnega grafa ?$G$?. Nazadnje, poveževa novi domnevi z že znanimi domnevami, ko dokaževa, da ?$S_{12}$?-domneva implicira ?$S_{10}$?-domnevo, in da ?$P_{12}$?-domneva ter ?$(5, 2)$?-ciklična krovna domneva skupaj implicirata ?$P_{10}$?-domnevo. Najino glavno orodje za dokaz zadnje trditve je nova reformulacija ?$(5, 2)$?-ciklične krovne domneve, ki pravi, da se povezavna množica poljubnega brezmostnega kubičpnega grafa brez trizobov da pokriti s štirimi popolnimi prirejanji"}],"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-5CVYESBI","edm:aggregatedCHO":{"@rdf:resource":"URN:NBN:SI:doc-5CVYESBI"},"edm:isShownBy":{"@rdf:resource":"http://www.dlib.si/stream/URN:NBN:SI:doc-5CVYESBI/dd640e36-b84e-455f-b575-7c5b5e176baa/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-5CVYESBI/maxi/edm"},"edm:isShownAt":{"@rdf:resource":"http://www.dlib.si/details/URN:NBN:SI:doc-5CVYESBI"}}}}