{"title":"边色完整图中的彩虹子图：回答厄尔多斯和图扎的两个问题","authors":"Maria Axenovich, Felix C. Clemen","doi":"10.1002/jgt.23063","DOIUrl":null,"url":null,"abstract":"<p>An edge-coloring of a complete graph with a set of colors <math>\n <semantics>\n <mrow>\n <mi>C</mi>\n </mrow>\n <annotation> $C$</annotation>\n </semantics></math> is called <i>completely balanced</i> if any vertex is incident to the same number of edges of each color from <math>\n <semantics>\n <mrow>\n <mi>C</mi>\n </mrow>\n <annotation> $C$</annotation>\n </semantics></math>. Erdős and Tuza asked in 1993 whether for any graph <math>\n <semantics>\n <mrow>\n <mi>F</mi>\n </mrow>\n <annotation> $F$</annotation>\n </semantics></math> on <math>\n <semantics>\n <mrow>\n <mi>ℓ</mi>\n </mrow>\n <annotation> $\\ell $</annotation>\n </semantics></math> edges and any completely balanced coloring of any sufficiently large complete graph using <math>\n <semantics>\n <mrow>\n <mi>ℓ</mi>\n </mrow>\n <annotation> $\\ell $</annotation>\n </semantics></math> colors contains a rainbow copy of <math>\n <semantics>\n <mrow>\n <mi>F</mi>\n </mrow>\n <annotation> $F$</annotation>\n </semantics></math>. This question was restated by Erdős in his list of “Some of my favourite problems on cycles and colourings.” We answer this question in the negative for most cliques <math>\n <semantics>\n <mrow>\n <mi>F</mi>\n <mo>=</mo>\n <msub>\n <mi>K</mi>\n <mi>q</mi>\n </msub>\n </mrow>\n <annotation> $F={K}_{q}$</annotation>\n </semantics></math> by giving explicit constructions of respective completely balanced colorings. Further, we answer a related question concerning completely balanced colorings of complete graphs with more colors than the number of edges in the graph <math>\n <semantics>\n <mrow>\n <mi>F</mi>\n </mrow>\n <annotation> $F$</annotation>\n </semantics></math>.</p>","PeriodicalId":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-12-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/jgt.23063","citationCount":"0","resultStr":"{\"title\":\"Rainbow subgraphs in edge-colored complete graphs: Answering two questions by Erdős and Tuza\",\"authors\":\"Maria Axenovich, Felix C. Clemen\",\"doi\":\"10.1002/jgt.23063\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>An edge-coloring of a complete graph with a set of colors <math>\\n <semantics>\\n <mrow>\\n <mi>C</mi>\\n </mrow>\\n <annotation> $C$</annotation>\\n </semantics></math> is called <i>completely balanced</i> if any vertex is incident to the same number of edges of each color from <math>\\n <semantics>\\n <mrow>\\n <mi>C</mi>\\n </mrow>\\n <annotation> $C$</annotation>\\n </semantics></math>. Erdős and Tuza asked in 1993 whether for any graph <math>\\n <semantics>\\n <mrow>\\n <mi>F</mi>\\n </mrow>\\n <annotation> $F$</annotation>\\n </semantics></math> on <math>\\n <semantics>\\n <mrow>\\n <mi>ℓ</mi>\\n </mrow>\\n <annotation> $\\\\ell $</annotation>\\n </semantics></math> edges and any completely balanced coloring of any sufficiently large complete graph using <math>\\n <semantics>\\n <mrow>\\n <mi>ℓ</mi>\\n </mrow>\\n <annotation> $\\\\ell $</annotation>\\n </semantics></math> colors contains a rainbow copy of <math>\\n <semantics>\\n <mrow>\\n <mi>F</mi>\\n </mrow>\\n <annotation> $F$</annotation>\\n </semantics></math>. This question was restated by Erdős in his list of “Some of my favourite problems on cycles and colourings.” We answer this question in the negative for most cliques <math>\\n <semantics>\\n <mrow>\\n <mi>F</mi>\\n <mo>=</mo>\\n <msub>\\n <mi>K</mi>\\n <mi>q</mi>\\n </msub>\\n </mrow>\\n <annotation> $F={K}_{q}$</annotation>\\n </semantics></math> by giving explicit constructions of respective completely balanced colorings. Further, we answer a related question concerning completely balanced colorings of complete graphs with more colors than the number of edges in the graph <math>\\n <semantics>\\n <mrow>\\n <mi>F</mi>\\n </mrow>\\n <annotation> $F$</annotation>\\n </semantics></math>.</p>\",\"PeriodicalId\":0,\"journal\":{\"name\":\"\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.0,\"publicationDate\":\"2023-12-12\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"https://onlinelibrary.wiley.com/doi/epdf/10.1002/jgt.23063\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://onlinelibrary.wiley.com/doi/10.1002/jgt.23063\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"","FirstCategoryId":"100","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1002/jgt.23063","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
Rainbow subgraphs in edge-colored complete graphs: Answering two questions by Erdős and Tuza
An edge-coloring of a complete graph with a set of colors is called completely balanced if any vertex is incident to the same number of edges of each color from . Erdős and Tuza asked in 1993 whether for any graph on edges and any completely balanced coloring of any sufficiently large complete graph using colors contains a rainbow copy of . This question was restated by Erdős in his list of “Some of my favourite problems on cycles and colourings.” We answer this question in the negative for most cliques by giving explicit constructions of respective completely balanced colorings. Further, we answer a related question concerning completely balanced colorings of complete graphs with more colors than the number of edges in the graph .
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