{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2023,9,13]],"date-time":"2023-09-13T05:13:41Z","timestamp":1694582021505},"reference-count":6,"publisher":"Cambridge University Press (CUP)","issue":"3","license":[{"start":{"date-parts":[[2013,2,28]],"date-time":"2013-02-28T00:00:00Z","timestamp":1362009600000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Combinator. Probab. Comp."],"published-print":{"date-parts":[[2013,5]]},"abstract":"<jats:p>For a positive integer <jats:italic>r<\/jats:italic> \u2265 2, a <jats:italic>K<jats:sub>r<\/jats:sub><\/jats:italic>-factor of a graph is a collection vertex-disjoint copies of <jats:italic>K<jats:sub>r<\/jats:sub><\/jats:italic> which covers all the vertices of the given graph. The celebrated theorem of Hajnal and Szemer\u00e9di asserts that every graph on <jats:italic>n<\/jats:italic> vertices with minimum degree at least <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0963548313000059_inline1\" \/><jats:tex-math>$(1-\\frac{1}{r})n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> contains a <jats:italic>K<jats:sub>r<\/jats:sub><\/jats:italic>-factor. In this note, we propose investigating the relation between minimum degree and existence of perfect <jats:italic>K<jats:sub>r<\/jats:sub><\/jats:italic>-packing for edge-weighted graphs. The main question we study is the following. Suppose that a positive integer <jats:italic>r<\/jats:italic> \u2265 2 and a real <jats:italic>t<\/jats:italic> \u2208 [0, 1] is given. What is the minimum weighted degree of <jats:italic>K<jats:sub>n<\/jats:sub><\/jats:italic> that guarantees the existence of a <jats:italic>K<jats:sub>r<\/jats:sub><\/jats:italic>-factor such that every factor has total edge weight at least <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0963548313000059_inline2\" \/><jats:tex-math>$$t\\binom{r}{2}$?$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> We provide some lower and upper bounds and make a conjecture on the asymptotics of the threshold as <jats:italic>n<\/jats:italic> goes to infinity.<\/jats:p>","DOI":"10.1017\/s0963548313000059","type":"journal-article","created":{"date-parts":[[2013,2,28]],"date-time":"2013-02-28T11:36:16Z","timestamp":1362051376000},"page":"346-350","source":"Crossref","is-referenced-by-count":1,"title":["Towards a Weighted Version of the Hajnal\u2013Szemer\u00e9di Theorem"],"prefix":"10.1017","volume":"22","author":[{"given":"JOZSEF","family":"BALOGH","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"GRAEME","family":"KEMKES","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"CHOONGBUM","family":"LEE","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"STEPHEN J.","family":"YOUNG","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2013,2,28]]},"reference":[{"key":"S0963548313000059_ref5","first-page":"601","volume-title":"Combinatorial Theory and its Applications II: Proc. Colloq., Balatonf\u00fcred, 1969","author":"Hajnal","year":"1970"},{"key":"S0963548313000059_ref4","doi-asserted-by":"publisher","DOI":"10.1112\/plms\/s3-2.1.69"},{"key":"S0963548313000059_ref1","unstructured":"Balogh J. , Kemkes G. , Lee C. and Young S. Towards a weighted version of the Hajnal\u2013Szemer\u00e9di theorem. arXiv:1206.1376 [math.CO]."},{"key":"S0963548313000059_ref3","doi-asserted-by":"publisher","DOI":"10.1017\/S0004972700006924"},{"key":"S0963548313000059_ref2","doi-asserted-by":"publisher","DOI":"10.1007\/BF01895727"},{"key":"S0963548313000059_ref6","doi-asserted-by":"publisher","DOI":"10.1137\/080729657"}],"container-title":["Combinatorics, Probability and Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0963548313000059","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,4,23]],"date-time":"2019-04-23T18:00:53Z","timestamp":1556042453000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0963548313000059\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2013,2,28]]},"references-count":6,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2013,5]]}},"alternative-id":["S0963548313000059"],"URL":"https:\/\/doi.org\/10.1017\/s0963548313000059","relation":{},"ISSN":["0963-5483","1469-2163"],"issn-type":[{"value":"0963-5483","type":"print"},{"value":"1469-2163","type":"electronic"}],"subject":[],"published":{"date-parts":[[2013,2,28]]}}}