{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,5,22]],"date-time":"2025-05-22T07:13:09Z","timestamp":1747897989202,"version":"3.40.5"},"reference-count":17,"publisher":"Wiley","license":[{"start":{"date-parts":[[2013,1,1]],"date-time":"2013-01-01T00:00:00Z","timestamp":1356998400000},"content-version":"unspecified","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by\/3.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["61271409","201001A151","2012M510768","2013T60264"],"award-info":[{"award-number":["61271409","201001A151","2012M510768","2013T60264"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Science and Technology Research and Development Program of Qinhuangdao","award":["61271409","201001A151","2012M510768","2013T60264"],"award-info":[{"award-number":["61271409","201001A151","2012M510768","2013T60264"]}]},{"DOI":"10.13039\/501100002858","name":"China Postdoctoral Science Foundation","doi-asserted-by":"publisher","award":["61271409","201001A151","2012M510768","2013T60264"],"award-info":[{"award-number":["61271409","201001A151","2012M510768","2013T60264"]}],"id":[{"id":"10.13039\/501100002858","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100002858","name":"China Postdoctoral Science Foundation","doi-asserted-by":"publisher","award":["61271409","201001A151","2012M510768","2013T60264"],"award-info":[{"award-number":["61271409","201001A151","2012M510768","2013T60264"]}],"id":[{"id":"10.13039\/501100002858","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Journal of Applied Mathematics"],"published-print":{"date-parts":[[2013]]},"abstract":"<jats:p>If a graph<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M2\"><mml:mrow><mml:mi>G<\/mml:mi><\/mml:mrow><\/mml:math>admits a<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M3\"><mml:mrow><mml:mi>k<\/mml:mi><\/mml:mrow><\/mml:math>-list assignment<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M4\"><mml:mrow><mml:mi>L<\/mml:mi><\/mml:mrow><\/mml:math>such that<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M5\"><mml:mrow><mml:mi>G<\/mml:mi><\/mml:mrow><\/mml:math>has a unique<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M6\"><mml:mrow><mml:mi>L<\/mml:mi><\/mml:mrow><\/mml:math>-coloring, then<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M7\"><mml:mrow><mml:mi>G<\/mml:mi><\/mml:mrow><\/mml:math>is called uniquely<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M8\"><mml:mrow><mml:mi>k<\/mml:mi><\/mml:mrow><\/mml:math>-list colorable graph, or U<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M9\"><mml:mrow><mml:mi>k<\/mml:mi><\/mml:mrow><\/mml:math>LC graph for short. In the process of characterizing U<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M10\"><mml:mrow><mml:mi>k<\/mml:mi><\/mml:mrow><\/mml:math>LC graphs, the complete multipartite graphs<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M11\"><mml:msub><mml:mrow><mml:mi>K<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn>1<\/mml:mn><mml:mi mathvariant=\"normal\">*<\/mml:mi><mml:mi>r<\/mml:mi><mml:mo>,<\/mml:mo><mml:mi>s<\/mml:mi><\/mml:mrow><\/mml:msub><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mi>r<\/mml:mi><mml:mo>,<\/mml:mo><mml:mi>s<\/mml:mi><mml:mo>\u2208<\/mml:mo><mml:mi>N<\/mml:mi><mml:mo stretchy=\"false\">)<\/mml:mo><\/mml:math>are often researched. But it is usually not easy to construct the unique<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M12\"><mml:mrow><mml:mi>k<\/mml:mi><\/mml:mrow><\/mml:math>-list assignment of<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M13\"><mml:mrow><mml:msub><mml:mrow><mml:mi>K<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn>1<\/mml:mn><mml:mi mathvariant=\"normal\">*<\/mml:mi><mml:mi>r<\/mml:mi><mml:mo>,<\/mml:mo><mml:mi>s<\/mml:mi><\/mml:mrow><\/mml:msub><\/mml:mrow><\/mml:math>. In this paper, we give some propositions about the property of the graph<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M14\"><mml:mrow><mml:msub><mml:mrow><mml:mi>K<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn>1<\/mml:mn><mml:mi mathvariant=\"normal\">*<\/mml:mi><mml:mi>r<\/mml:mi><mml:mo>,<\/mml:mo><mml:mi>s<\/mml:mi><\/mml:mrow><\/mml:msub><\/mml:mrow><\/mml:math>when it is U<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M15\"><mml:mrow><mml:mi>k<\/mml:mi><\/mml:mrow><\/mml:math>LC, which provide a very significant guide for constructing such list assignment. Then a special example of U<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M16\"><mml:mrow><mml:mi>k<\/mml:mi><\/mml:mrow><\/mml:math>LC graphs<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M17\"><mml:mrow><mml:msub><mml:mrow><mml:mi>K<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn>1<\/mml:mn><mml:mi mathvariant=\"normal\">*<\/mml:mi><mml:mi>r<\/mml:mi><mml:mo>,<\/mml:mo><mml:mi>s<\/mml:mi><\/mml:mrow><\/mml:msub><\/mml:mrow><\/mml:math>as a application of these propositions is introduced. The conclusion will pave the way to characterize U<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M18\"><mml:mrow><mml:mi>k<\/mml:mi><\/mml:mrow><\/mml:math>LC complete multipartite graphs.<\/jats:p>","DOI":"10.1155\/2013\/380861","type":"journal-article","created":{"date-parts":[[2013,10,8]],"date-time":"2013-10-08T17:08:44Z","timestamp":1381252124000},"page":"1-5","source":"Crossref","is-referenced-by-count":2,"title":["Some Conclusion on Unique<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M1\"><mml:mrow><mml:mi>k<\/mml:mi><\/mml:mrow><\/mml:math>-List Colorable Complete Multipartite Graphs"],"prefix":"10.1155","volume":"2013","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-4181-9528","authenticated-orcid":true,"given":"Yanning","family":"Wang","sequence":"first","affiliation":[{"name":"School of Sciences, Yanshan University, Qinhuangdao, Hebei 066004, China"},{"name":"Key Lab of Industrial Computer Control Engineering of Hebei Province, Institute of Electrical Engineering, Yanshan University, Qinhuangdao, Hebei 066004, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9730-6871","authenticated-orcid":true,"given":"Yanyan","family":"Wang","sequence":"additional","affiliation":[{"name":"School of Economics and Management, Yanshan University, Qinhuangdao, Hebei 066004, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Xuguang","family":"Zhang","sequence":"additional","affiliation":[{"name":"Key Lab of Industrial Computer Control Engineering of Hebei Province, Institute of Electrical Engineering, Yanshan University, Qinhuangdao, Hebei 066004, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"311","reference":[{"year":"2001","key":"1"},{"key":"2","first-page":"151","volume":"41","year":"2002","journal-title":"Journal of Combinatorial Mathematics and Combinatorial Computing"},{"issue":"29","key":"3","first-page":"3","volume":"101","year":"1976","journal-title":"Diskretnyi Analiz"},{"key":"5","first-page":"295","volume":"51","year":"1999","journal-title":"Ars Combinatoria"},{"issue":"3","key":"7","doi-asserted-by":"crossref","first-page":"479","DOI":"10.3724\/SP.J.1001.2008.00479","volume":"19","year":"2008","journal-title":"Journal of Software"},{"issue":"4","key":"8","doi-asserted-by":"crossref","first-page":"555","DOI":"10.1007\/s11036-006-7322-y","volume":"11","year":"2006","journal-title":"Mobile Networks and Applications"},{"key":"9","doi-asserted-by":"publisher","DOI":"10.1016\/j.jctb.2005.12.001"},{"key":"10","first-page":"105","volume":"11","year":"1995","journal-title":"The Australasian Journal of Combinatorics"},{"key":"12","doi-asserted-by":"publisher","DOI":"10.1016\/S0012-365X(96)00247-6"},{"key":"13","first-page":"231","volume":"113","year":"1996","journal-title":"Congressus Numerantium"},{"key":"14","first-page":"307","volume":"59","year":"2001","journal-title":"Ars Combinatoria"},{"key":"15","doi-asserted-by":"publisher","DOI":"10.1016\/j.tcs.2008.03.018"},{"key":"16","first-page":"367","volume":"88","year":"2008","journal-title":"Ars Combinatoria"},{"issue":"2","key":"17","first-page":"211","volume":"35","year":"2006","journal-title":"The Australasian Journal of Combinatorics"},{"key":"18","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-540-70666-3_30"},{"key":"19","first-page":"203","volume":"93","year":"2009","journal-title":"Ars Combinatoria"},{"key":"20"}],"container-title":["Journal of Applied Mathematics"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/downloads.hindawi.com\/journals\/jam\/2013\/380861.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/downloads.hindawi.com\/journals\/jam\/2013\/380861.xml","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/downloads.hindawi.com\/journals\/jam\/2013\/380861.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2017,6,21]],"date-time":"2017-06-21T19:22:53Z","timestamp":1498072973000},"score":1,"resource":{"primary":{"URL":"http:\/\/www.hindawi.com\/journals\/jam\/2013\/380861\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2013]]},"references-count":17,"alternative-id":["380861","380861"],"URL":"https:\/\/doi.org\/10.1155\/2013\/380861","relation":{},"ISSN":["1110-757X","1687-0042"],"issn-type":[{"type":"print","value":"1110-757X"},{"type":"electronic","value":"1687-0042"}],"subject":[],"published":{"date-parts":[[2013]]}}}