{"?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-MTYTYNX7/75c7fdc0-4aee-4dbf-9c4d-83d6218c18c8/PDF","dcterms:extent":"3090 KB"},{"@rdf:about":"http://www.dlib.si/stream/URN:NBN:SI:doc-MTYTYNX7/d49a7471-a817-4891-ae08-ccbf2c2e9063/TEXT","dcterms:extent":"111 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-MTYTYNX7","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":["Cavers, Michael","Seyffarth, Karen"],"dc:format":[{"@xml:lang":"sl","#text":"letnik:17"},{"@xml:lang":"sl","#text":"številka:2"},{"@xml:lang":"sl","#text":"str. 653-698"}],"dc:identifier":["ISSN:1855-3966","COBISSID_HOST:18976857","URN:URN:NBN:SI:doc-MTYTYNX7"],"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":"2-drevesa"},{"@xml:lang":"en","#text":"2-trees"},{"@xml:lang":"sl","#text":"barvanje grafa"},{"@xml:lang":"en","#text":"graph colouring"},{"@xml:lang":"en","#text":"Gray codes"},{"@xml:lang":"sl","#text":"Grayeve kode"},{"@xml:lang":"en","#text":"Hamilton cycles"},{"@xml:lang":"sl","#text":"hamiltonski cikli"},{"@xml:lang":"en","#text":"reconfiguration problems"},{"@xml:lang":"sl","#text":"rekonfiguracijski problemi"}],"dcterms:temporal":{"@rdf:resource":"2008-2025"},"dc:title":{"@xml:lang":"sl","#text":"Reconfiguring vertex colourings of 2-trees|"},"dc:description":[{"@xml:lang":"sl","#text":"Let ?$H$? be a graph and let ?$k \\geq \\chi (H)$? be an integer. The ?$k$?-colouring graph of ?$H$?, denoted ?$G_k(H)$?, is the graph whose vertex set consists of all proper ?$k$?-vertex-colourings (or simply ?$k$?-colourings) of ?$H$? using colours ?$\\{1, 2, \\dots, k\\}$?; two vertices of ?$G_k(H)$? are adjacent if and only if the corresponding ?$k$?-colourings differ in colour on exactly one vertex of ?$H$?. If ?$G_k(H)$? has a Hamilton cycle, then ?$H$? is said to have a Gray code of ?$k$?-colourings, and the Gray code number of ?$H$? is the least integer ?$k_0(H)$? such that ?$G_k(H)$? has a Gray code of ?$k$?-colourings for all ?$k \\geq k_0(H)$?. K. Choo and G. MacGillivray determine the Gray code numbers of trees. We extend this result to 2-trees. A 2-tree is constructed recursively by starting with a complete graph on three vertices and connecting each new vertex to an existing clique on two vertices. We prove that if ?$H$? is a 2-tree, then ?$k_0(H) = 4$? unless ?$H$? is isomorphic to the join of a tree ?$T$? and a vertex ?$u$?, where ?$T$? is a star on at least three vertices, or the bipartition of ?$T$? has two even parts; in these cases, ?$k_0(H) = 5$?"},{"@xml:lang":"sl","#text":"Naj bo ?$H$? graf in naj bo ?$k \\geq \\chi (H)$? celo število. Graf ?$k?$-barvanj grafa ?$H$?, označen z ?$G_k(H)$?, je graf, katerega množica vozlišč sestoji iz vseh pravih ?$k$?-vozliščnih barvanj (ali preprosto ?$k$?-barvanj) grafa ?$H$? z barvami ?$\\{1, 2, \\dots, k\\}$?; vozlišči grafa ?$G_k(H)$? sta sosednji, če in samočce se ustrezni ?$k$?-barvanji razlikujeta v barvi natanko enega vozlišča grafa ?$H$?. Če ima ?$G_k(H)$? hamiltonski cikel, potem pravimo, da ima ?$H$? Grayevo kodo iz ?$k$?-barvanj. Najmanjše celo število ?$k_0(H)$?, pri katerem ima ?$G_k(H)$? Grayevo kodo iz ?$k$?-barvanj za vse ?$k \\geq k_0(H)$?, imenujemo prag Grayeve kode grafa ?$H$?. Choo in MacGillivray sta določila pragove Grayevih kod za drevesa. V pričujočem članku razširimo ta rezultat na 2-drevesa. Konstrukcija 2-drevesa poteka rekurzivno: začnemo s polnim grafom na treh vozliščih, potem pa vsako novo vozlišče dodamo neki že obstoječi kliki na dveh vozliščih. Dokažemo, da če je ?$H$? 2-drevo, potem je ?$k_0(H) = 4$?, razen če je ?$H$? izomorfen spoju drevesa ?$T$? in vozlišča ?$u$?, kjer je ?$T$? zvezda na najmanj treh vozlišcih, ali če se ?$T$? deli na dva enako velika dela; v teh dveh primerih je ?$k_0(H) = 5$?"}],"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-MTYTYNX7","edm:aggregatedCHO":{"@rdf:resource":"URN:NBN:SI:doc-MTYTYNX7"},"edm:isShownBy":{"@rdf:resource":"http://www.dlib.si/stream/URN:NBN:SI:doc-MTYTYNX7/75c7fdc0-4aee-4dbf-9c4d-83d6218c18c8/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-MTYTYNX7/maxi/edm"},"edm:isShownAt":{"@rdf:resource":"http://www.dlib.si/details/URN:NBN:SI:doc-MTYTYNX7"}}}}