{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,2,21]],"date-time":"2025-02-21T07:21:01Z","timestamp":1740122461170,"version":"3.37.3"},"reference-count":25,"publisher":"Springer Science and Business Media LLC","issue":"2","license":[{"start":{"date-parts":[[2018,5,17]],"date-time":"2018-05-17T00:00:00Z","timestamp":1526515200000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"funder":[{"DOI":"10.13039\/501100001809","name":"NSFC","doi-asserted-by":"crossref","award":["11531011"],"award-info":[{"award-number":["11531011"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J Comb Optim"],"published-print":{"date-parts":[[2018,8]]},"DOI":"10.1007\/s10878-018-0307-5","type":"journal-article","created":{"date-parts":[[2018,5,17]],"date-time":"2018-05-17T14:19:31Z","timestamp":1526566771000},"page":"458-471","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["The $$(k,\\ell )$$ ( k , \u2113 ) -proper index of graphs"],"prefix":"10.1007","volume":"36","author":[{"given":"Hong","family":"Chang","sequence":"first","affiliation":[]},{"given":"Xueliang","family":"Li","sequence":"additional","affiliation":[]},{"given":"Colton","family":"Magnant","sequence":"additional","affiliation":[]},{"given":"Zhongmei","family":"Qin","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2018,5,17]]},"reference":[{"key":"307_CR1","volume-title":"The probabilistic method","author":"N Alon","year":"2004","unstructured":"Alon N, Spencer JH (2004) The probabilistic method. Wiley, Hoboken"},{"key":"307_CR2","first-page":"189","volume":"97","author":"E Andrews","year":"2016","unstructured":"Andrews E, Laforge E, Lumduanhom C, Zhang P (2016) On proper-path colorings in graphs. J Comb Math Comb Comput 97:189\u2013207","journal-title":"J Comb Math Comb Comput"},{"key":"307_CR3","volume-title":"Extremal graph theory, London mathematical society monographs","author":"B Bollob\u00e1s","year":"1978","unstructured":"Bollob\u00e1s B (1978) Extremal graph theory, London mathematical society monographs, vol 11. Academic Press Inc, London"},{"key":"307_CR4","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-84628-970-5","volume-title":"Graph theory, GTM","author":"JA Bondy","year":"2008","unstructured":"Bondy JA, Murty USR (2008) Graph theory, GTM, vol 244. Springer, Berlin"},{"key":"307_CR5","doi-asserted-by":"publisher","first-page":"2550","DOI":"10.1016\/j.disc.2011.09.003","volume":"312","author":"V Borozan","year":"2012","unstructured":"Borozan V, Fujita S, Gerek A, Magnant C, Manoussakis Y, Montero L, Tuza Z (2012) Proper connection of graphs. Discrete Math 312:2550\u20132560","journal-title":"Discrete Math"},{"key":"307_CR6","doi-asserted-by":"publisher","first-page":"220","DOI":"10.1002\/net.21513","volume":"62","author":"Q Cai","year":"2013","unstructured":"Cai Q, Li X, Song J (2013) Solutions to conjectures on $$(k,\\ell )$$ ( k , \u2113 ) -rainbow index of complete graphs. Networks 62:220\u2013224","journal-title":"Networks"},{"issue":"3","key":"307_CR7","doi-asserted-by":"publisher","first-page":"1225","DOI":"10.1007\/s40840-016-0348-9","volume":"39","author":"Q Cai","year":"2016","unstructured":"Cai Q, Li X, Song J (2016) The $$(k,\\ell )$$ ( k , \u2113 ) -rainbow index of complete bipartite and multipartitegraphs. Bull Malays Math Sci Soc 39(3):1225\u20131236","journal-title":"Bull Malays Math Sci Soc"},{"issue":"2","key":"307_CR8","doi-asserted-by":"publisher","first-page":"765","DOI":"10.1007\/s40840-015-0301-3","volume":"39","author":"Q Cai","year":"2016","unstructured":"Cai Q, Li X, Song J (2016) The $$(k,\\ell )$$ ( k , \u2113 ) -rainbow index of random graphs. Bull Malays Math Sci Soc 39(2):765\u2013771","journal-title":"Bull Malays Math Sci Soc"},{"issue":"2","key":"307_CR9","doi-asserted-by":"publisher","first-page":"1142","DOI":"10.1007\/s10878-014-9815-0","volume":"31","author":"Q Cai","year":"2016","unstructured":"Cai Q, Li X, Zhao Y (2016) The 3-rainbow index and connected dominating sets. J Comb Optim 31(2):1142\u20131159","journal-title":"J Comb Optim"},{"key":"307_CR10","doi-asserted-by":"publisher","first-page":"68","DOI":"10.1016\/j.dam.2015.10.019","volume":"209","author":"Q Cai","year":"2016","unstructured":"Cai Q, Li X, Zhao Y (2016) Note on the upper bound of the rianbow index of a graph. Discrete Appl Math 209:68\u201374","journal-title":"Discrete Appl Math"},{"key":"307_CR11","doi-asserted-by":"publisher","DOI":"10.1007\/s40840-016-0404-5","author":"H Chang","year":"2016","unstructured":"Chang H, Li X, Qin Z (2016) Some upper bounds for the 3-proper index of graphs. Bull Malays Math Sci Soc. https:\/\/doi.org\/10.1007\/s40840-016-0404-5","journal-title":"Bull Malays Math Sci Soc"},{"key":"307_CR12","doi-asserted-by":"publisher","first-page":"75","DOI":"10.1002\/net.20296","volume":"54","author":"G Chartrand","year":"2009","unstructured":"Chartrand G, Johns GL, McKeon KA, Zhang P (2009) The rainbow connectivity of a graph. Networks 54:75\u201381","journal-title":"Networks"},{"key":"307_CR13","first-page":"360","volume":"55","author":"G Chartrand","year":"2010","unstructured":"Chartrand G, Okamoto F, Zhang P (2010) Rainbow trees in graphs and generalized connectivity. Networks 55:360\u2013367","journal-title":"Networks"},{"issue":"1","key":"307_CR14","doi-asserted-by":"crossref","first-page":"57","DOI":"10.1016\/j.amc.2016.10.025","volume":"296","author":"L Chen","year":"2017","unstructured":"Chen L, Li X, Liu J (2017) The $$k$$ k -proper index of graphs. Appl Math Comput 296(1):57\u201363","journal-title":"Appl Math Comput"},{"key":"307_CR15","doi-asserted-by":"publisher","first-page":"81","DOI":"10.7151\/dmgt.1780","volume":"35","author":"L Chen","year":"2015","unstructured":"Chen L, Li X, Yang K, Zhao Y (2015) The 3-rainbow index of a graph. Discuss Math Graph Theory 35:81\u201394","journal-title":"Discuss Math Graph Theory"},{"key":"307_CR16","first-page":"17","volume":"5","author":"P Erd\u0151s","year":"1960","unstructured":"Erd\u0151s P, R\u00e9nyi A (1960) On the evolution of random graphs. Publ Math Inst Hungar Acad Sci 5:17\u201361","journal-title":"Publ Math Inst Hungar Acad Sci"},{"issue":"2","key":"307_CR17","doi-asserted-by":"publisher","first-page":"336","DOI":"10.1016\/j.tcs.2015.10.017","volume":"609","author":"R Gu","year":"2016","unstructured":"Gu R, Li X, Qin Z (2016) Proper connection number of random graphs. Theor Comput Sci 609(2):336\u2013343","journal-title":"Theor Comput Sci"},{"key":"307_CR18","volume-title":"Random graphs, Wiley-Interscience series in discrete mathematics and optimization","author":"S Janson","year":"2000","unstructured":"Janson S, Luczak T, Ruci\u0144ski A (2000) Random graphs, Wiley-Interscience series in discrete mathematics and optimization. Wiley, New York"},{"key":"307_CR19","doi-asserted-by":"crossref","unstructured":"Li X, Magnant C (2015) Properly colored notions of connectivity\u2013a dynamic survey. Theory Appl Graphs. https:\/\/doi.org\/10.20429\/tag.2015.000102","DOI":"10.20429\/tag.2015.000102"},{"key":"307_CR20","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-319-89617-5","volume-title":"Properly colored connectivity of graphs. Springer briefs in math","author":"X Li","year":"2018","unstructured":"Li X, Magnant C, Qin Z (2018) Properly colored connectivity of graphs. Springer briefs in math. Springer, Cham"},{"key":"307_CR21","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-319-33828-6","volume-title":"Generalized connectivity of graphs. Springer briefs in math","author":"X Li","year":"2016","unstructured":"Li X, Mao Y (2016) Generalized connectivity of graphs. Springer briefs in math. Springer, New York"},{"key":"307_CR22","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/s00373-012-1243-2","volume":"29","author":"X Li","year":"2013","unstructured":"Li X, Shi Y, Sun Y (2013) Rainbow connections of graphs: a survey. Graphs Combin 29:1\u201338","journal-title":"Graphs Combin"},{"key":"307_CR23","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4614-3119-0","volume-title":"Rainbow connections of graphs. Springer briefs in math","author":"X Li","year":"2012","unstructured":"Li X, Sun Y (2012) Rainbow connections of graphs. Springer briefs in math. Springer, New York"},{"key":"307_CR24","doi-asserted-by":"publisher","first-page":"480","DOI":"10.1016\/j.tcs.2015.06.006","volume":"607","author":"X Li","year":"2015","unstructured":"Li X, Wei M, Yue J (2015) Proper connection number and connected dominating sets. Theor Comput Sci 607:480\u2013487","journal-title":"Theor Comput Sci"},{"key":"307_CR25","unstructured":"Sauer N (1967) Extremaleigenschaften regul\u00e4rer graphen gegebener taillenweite I and II. Sitzungsberichte \u00d6sterreich Akad Wiss Math Natur Kl, S-B II 176: 9\u201325, 27\u201343"}],"container-title":["Journal of Combinatorial Optimization"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s10878-018-0307-5\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10878-018-0307-5.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10878-018-0307-5.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,11,2]],"date-time":"2020-11-02T03:27:17Z","timestamp":1604287637000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s10878-018-0307-5"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,5,17]]},"references-count":25,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2018,8]]}},"alternative-id":["307"],"URL":"https:\/\/doi.org\/10.1007\/s10878-018-0307-5","relation":{},"ISSN":["1382-6905","1573-2886"],"issn-type":[{"type":"print","value":"1382-6905"},{"type":"electronic","value":"1573-2886"}],"subject":[],"published":{"date-parts":[[2018,5,17]]},"assertion":[{"value":"17 May 2018","order":1,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}