Stijn Cambie, Wouter Cames van Batenburg, Ewan Davies, Ross J. Kang
{"title":"有界阶数图的包装数列表","authors":"Stijn Cambie, Wouter Cames van Batenburg, Ewan Davies, Ross J. Kang","doi":"10.1017/s0963548324000191","DOIUrl":null,"url":null,"abstract":"We investigate the list packing number of a graph, the least <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548324000191_inline1.png\"/> <jats:tex-math> $k$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> such that there are always <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548324000191_inline2.png\"/> <jats:tex-math> $k$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> disjoint proper list-colourings whenever we have lists all of size <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548324000191_inline3.png\"/> <jats:tex-math> $k$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> associated to the vertices. We are curious how the behaviour of the list packing number contrasts with that of the list chromatic number, particularly in the context of bounded degree graphs. The main question we pursue is whether every graph with maximum degree <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548324000191_inline4.png\"/> <jats:tex-math> $\\Delta$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> has list packing number at most <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548324000191_inline5.png\"/> <jats:tex-math> $\\Delta +1$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>. Our results highlight the subtleties of list packing and the barriers to, for example, pursuing a Brooks’-type theorem for the list packing number.","PeriodicalId":10503,"journal":{"name":"Combinatorics, Probability and Computing","volume":null,"pages":null},"PeriodicalIF":0.0000,"publicationDate":"2024-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"List packing number of bounded degree graphs\",\"authors\":\"Stijn Cambie, Wouter Cames van Batenburg, Ewan Davies, Ross J. Kang\",\"doi\":\"10.1017/s0963548324000191\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"We investigate the list packing number of a graph, the least <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" mime-subtype=\\\"png\\\" xlink:href=\\\"S0963548324000191_inline1.png\\\"/> <jats:tex-math> $k$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> such that there are always <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" mime-subtype=\\\"png\\\" xlink:href=\\\"S0963548324000191_inline2.png\\\"/> <jats:tex-math> $k$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> disjoint proper list-colourings whenever we have lists all of size <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" mime-subtype=\\\"png\\\" xlink:href=\\\"S0963548324000191_inline3.png\\\"/> <jats:tex-math> $k$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> associated to the vertices. We are curious how the behaviour of the list packing number contrasts with that of the list chromatic number, particularly in the context of bounded degree graphs. The main question we pursue is whether every graph with maximum degree <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" mime-subtype=\\\"png\\\" xlink:href=\\\"S0963548324000191_inline4.png\\\"/> <jats:tex-math> $\\\\Delta$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> has list packing number at most <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" mime-subtype=\\\"png\\\" xlink:href=\\\"S0963548324000191_inline5.png\\\"/> <jats:tex-math> $\\\\Delta +1$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>. Our results highlight the subtleties of list packing and the barriers to, for example, pursuing a Brooks’-type theorem for the list packing number.\",\"PeriodicalId\":10503,\"journal\":{\"name\":\"Combinatorics, Probability and Computing\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2024-09-18\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Combinatorics, Probability and Computing\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.1017/s0963548324000191\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Combinatorics, Probability and Computing","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1017/s0963548324000191","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
We investigate the list packing number of a graph, the least $k$ such that there are always $k$ disjoint proper list-colourings whenever we have lists all of size $k$ associated to the vertices. We are curious how the behaviour of the list packing number contrasts with that of the list chromatic number, particularly in the context of bounded degree graphs. The main question we pursue is whether every graph with maximum degree $\Delta$ has list packing number at most $\Delta +1$ . Our results highlight the subtleties of list packing and the barriers to, for example, pursuing a Brooks’-type theorem for the list packing number.
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