{"title":"针对所有 $$k>4$$ 的列表-k-着色无 H 图","authors":"Maria Chudnovsky, Sepehr Hajebi, Sophie Spirkl","doi":"10.1007/s00493-024-00106-2","DOIUrl":null,"url":null,"abstract":"<p>Given an integer <span>\\(k>4\\)</span> and a graph <i>H</i>, we prove that, assuming <span>P</span><span>\\(\\ne \\)</span> <span>NP</span>, the <span>List-</span><i>k</i> <span>-Coloring Problem</span> restricted to <i>H</i>-free graphs can be solved in polynomial time if and only if either every component of <i>H</i> is a path on at most three vertices, or removing the isolated vertices of <i>H</i> leaves an induced subgraph of the five-vertex path. In fact, the “if” implication holds for all <span>\\(k\\ge 1\\)</span>.\n</p>","PeriodicalId":50666,"journal":{"name":"Combinatorica","volume":"47 1","pages":""},"PeriodicalIF":1.0000,"publicationDate":"2024-05-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"List-k-Coloring H-Free Graphs for All $$k>4$$\",\"authors\":\"Maria Chudnovsky, Sepehr Hajebi, Sophie Spirkl\",\"doi\":\"10.1007/s00493-024-00106-2\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>Given an integer <span>\\\\(k>4\\\\)</span> and a graph <i>H</i>, we prove that, assuming <span>P</span><span>\\\\(\\\\ne \\\\)</span> <span>NP</span>, the <span>List-</span><i>k</i> <span>-Coloring Problem</span> restricted to <i>H</i>-free graphs can be solved in polynomial time if and only if either every component of <i>H</i> is a path on at most three vertices, or removing the isolated vertices of <i>H</i> leaves an induced subgraph of the five-vertex path. In fact, the “if” implication holds for all <span>\\\\(k\\\\ge 1\\\\)</span>.\\n</p>\",\"PeriodicalId\":50666,\"journal\":{\"name\":\"Combinatorica\",\"volume\":\"47 1\",\"pages\":\"\"},\"PeriodicalIF\":1.0000,\"publicationDate\":\"2024-05-14\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Combinatorica\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1007/s00493-024-00106-2\",\"RegionNum\":2,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Combinatorica","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s00493-024-00106-2","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
Given an integer \(k>4\) and a graph H, we prove that, assuming P\(\ne \)NP, the List-k-Coloring Problem restricted to H-free graphs can be solved in polynomial time if and only if either every component of H is a path on at most three vertices, or removing the isolated vertices of H leaves an induced subgraph of the five-vertex path. In fact, the “if” implication holds for all \(k\ge 1\).
期刊介绍：
COMBINATORICA publishes research papers in English in a variety of areas of combinatorics and the theory of computing, with particular emphasis on general techniques and unifying principles. Typical but not exclusive topics covered by COMBINATORICA are
- Combinatorial structures (graphs, hypergraphs, matroids, designs, permutation groups).
- Combinatorial optimization.
- Combinatorial aspects of geometry and number theory.
- Algorithms in combinatorics and related fields.
- Computational complexity theory.
- Randomization and explicit construction in combinatorics and algorithms.
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