具有彩虹 C4 ${C}_{4}$-s 的平面图的适当边着色

IF 0.9 3区数学 Q2 MATHEMATICS

Journal of Graph Theory Pub Date : 2024-08-05 DOI:10.1002/jgt.23163

András Gyárfás, Ryan R. Martin, Miklós Ruszinkó, Gábor N. Sárközy

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Motivated by earlier papers on B-colorings, here we consider <math>\n <semantics>\n <mrow>\n \n <mrow>\n <msub>\n <mi>q</mi>\n \n <mi>B</mi>\n </msub>\n \n <mrow>\n <mo>(</mo>\n \n <mi>G</mi>\n \n <mo>)</mo>\n </mrow>\n </mrow>\n </mrow>\n <annotation> ${q}_{B}(G)$</annotation>\n </semantics></math> for planar and outerplanar graphs in terms of the maximum degree <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>Δ</mi>\n \n <mo>=</mo>\n \n <mi>Δ</mi>\n \n <mrow>\n <mo>(</mo>\n \n <mi>G</mi>\n \n <mo>)</mo>\n </mrow>\n </mrow>\n </mrow>\n <annotation> ${\\rm{\\Delta }}={\\rm{\\Delta }}(G)$</annotation>\n </semantics></math>. We prove that <math>\n <semantics>\n <mrow>\n \n <mrow>\n <msub>\n <mi>q</mi>\n \n <mi>B</mi>\n </msub>\n \n <mrow>\n <mo>(</mo>\n \n <mi>G</mi>\n \n <mo>)</mo>\n </mrow>\n \n <mo>≤</mo>\n \n <mn>2</mn>\n \n <mi>Δ</mi>\n \n <mo>+</mo>\n \n <mn>8</mn>\n </mrow>\n </mrow>\n <annotation> ${q}_{B}(G)\\le 2{\\rm{\\Delta }}+8$</annotation>\n </semantics></math> for planar graphs, <math>\n <semantics>\n <mrow>\n \n <mrow>\n <msub>\n <mi>q</mi>\n \n <mi>B</mi>\n </msub>\n \n <mrow>\n <mo>(</mo>\n \n <mi>G</mi>\n \n <mo>)</mo>\n </mrow>\n \n <mo>≤</mo>\n \n <mn>2</mn>\n \n <mi>Δ</mi>\n </mrow>\n </mrow>\n <annotation> ${q}_{B}(G)\\le 2{\\rm{\\Delta }}$</annotation>\n </semantics></math> for bipartite planar graphs, and <math>\n <semantics>\n <mrow>\n \n <mrow>\n <msub>\n <mi>q</mi>\n \n <mi>B</mi>\n </msub>\n \n <mrow>\n <mo>(</mo>\n \n <mi>G</mi>\n \n <mo>)</mo>\n </mrow>\n \n <mo>≤</mo>\n \n <mi>Δ</mi>\n \n <mo>+</mo>\n \n <mn>1</mn>\n </mrow>\n </mrow>\n <annotation> ${q}_{B}(G)\\le {\\rm{\\Delta }}+1$</annotation>\n </semantics></math> for outerplanar graphs with <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>Δ</mi>\n \n <mo>≥</mo>\n \n <mn>4</mn>\n </mrow>\n </mrow>\n <annotation> ${\\rm{\\Delta }}\\ge 4$</annotation>\n </semantics></math>. We conjecture that, for <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>Δ</mi>\n </mrow>\n </mrow>\n <annotation> ${\\rm{\\Delta }}$</annotation>\n </semantics></math> sufficiently large, <math>\n <semantics>\n <mrow>\n \n <mrow>\n <msub>\n <mi>q</mi>\n \n <mi>B</mi>\n </msub>\n \n <mrow>\n <mo>(</mo>\n \n <mi>G</mi>\n \n <mo>)</mo>\n </mrow>\n \n <mo>≤</mo>\n \n <mn>2</mn>\n \n <mi>Δ</mi>\n \n <mrow>\n <mo>(</mo>\n \n <mi>G</mi>\n \n <mo>)</mo>\n </mrow>\n </mrow>\n </mrow>\n <annotation> ${q}_{B}(G)\\le 2{\\rm{\\Delta }}(G)$</annotation>\n </semantics></math> for planar <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>G</mi>\n </mrow>\n </mrow>\n <annotation> $G$</annotation>\n </semantics></math>, and <math>\n <semantics>\n <mrow>\n \n <mrow>\n <msub>\n <mi>q</mi>\n \n <mi>B</mi>\n </msub>\n \n <mrow>\n <mo>(</mo>\n \n <mi>G</mi>\n \n <mo>)</mo>\n </mrow>\n \n <mo>≤</mo>\n \n <mi>Δ</mi>\n \n <mrow>\n <mo>(</mo>\n \n <mi>G</mi>\n \n <mo>)</mo>\n </mrow>\n </mrow>\n </mrow>\n <annotation> ${q}_{B}(G)\\le {\\rm{\\Delta }}(G)$</annotation>\n </semantics></math> for outerplanar <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>G</mi>\n </mrow>\n </mrow>\n <annotation> $G$</annotation>\n </semantics></math>.","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"107 4","pages":"833-846"},"PeriodicalIF":0.9000,"publicationDate":"2024-08-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Proper edge colorings of planar graphs with rainbow \\n \\n \\n \\n \\n C\\n 4\\n \\n \\n \\n ${C}_{4}$\\n -s\",\"authors\":\"András Gyárfás, Ryan R. Martin, Miklós Ruszinkó, Gábor N. Sárközy\",\"doi\":\"10.1002/jgt.23163\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"We call a proper edge coloring of a graph <math>\\n <semantics>\\n <mrow>\\n \\n <mrow>\\n <mi>G</mi>\\n </mrow>\\n </mrow>\\n <annotation> $G$</annotation>\\n </semantics></math> a B-coloring if every 4-cycle of <math>\\n <semantics>\\n <mrow>\\n \\n <mrow>\\n <mi>G</mi>\\n </mrow>\\n </mrow>\\n <annotation> $G$</annotation>\\n </semantics></math> is colored with four different colors. Let <math>\\n <semantics>\\n <mrow>\\n \\n <mrow>\\n <msub>\\n <mi>q</mi>\\n \\n <mi>B</mi>\\n </msub>\\n \\n <mrow>\\n <mo>(</mo>\\n \\n <mi>G</mi>\\n \\n <mo>)</mo>\\n </mrow>\\n </mrow>\\n </mrow>\\n <annotation> ${q}_{B}(G)$</annotation>\\n </semantics></math> denote the smallest number of colors needed for a B-coloring of <math>\\n <semantics>\\n <mrow>\\n \\n <mrow>\\n <mi>G</mi>\\n </mrow>\\n </mrow>\\n <annotation> $G$</annotation>\\n </semantics></math>. Motivated by earlier papers on B-colorings, here we consider <math>\\n <semantics>\\n <mrow>\\n \\n <mrow>\\n <msub>\\n <mi>q</mi>\\n \\n <mi>B</mi>\\n </msub>\\n \\n <mrow>\\n <mo>(</mo>\\n \\n <mi>G</mi>\\n \\n <mo>)</mo>\\n </mrow>\\n </mrow>\\n </mrow>\\n <annotation> ${q}_{B}(G)$</annotation>\\n </semantics></math> for planar and outerplanar graphs in terms of the maximum degree <math>\\n <semantics>\\n <mrow>\\n \\n <mrow>\\n <mi>Δ</mi>\\n \\n <mo>=</mo>\\n \\n <mi>Δ</mi>\\n \\n <mrow>\\n <mo>(</mo>\\n \\n <mi>G</mi>\\n \\n <mo>)</mo>\\n </mrow>\\n </mrow>\\n </mrow>\\n <annotation> ${\\\\rm{\\\\Delta }}={\\\\rm{\\\\Delta }}(G)$</annotation>\\n </semantics></math>. We prove that <math>\\n <semantics>\\n <mrow>\\n \\n <mrow>\\n <msub>\\n <mi>q</mi>\\n \\n <mi>B</mi>\\n </msub>\\n \\n <mrow>\\n <mo>(</mo>\\n \\n <mi>G</mi>\\n \\n <mo>)</mo>\\n </mrow>\\n \\n <mo>≤</mo>\\n \\n <mn>2</mn>\\n \\n <mi>Δ</mi>\\n \\n <mo>+</mo>\\n \\n <mn>8</mn>\\n </mrow>\\n </mrow>\\n <annotation> ${q}_{B}(G)\\\\le 2{\\\\rm{\\\\Delta }}+8$</annotation>\\n </semantics></math> for planar graphs, <math>\\n <semantics>\\n <mrow>\\n \\n <mrow>\\n <msub>\\n <mi>q</mi>\\n \\n <mi>B</mi>\\n </msub>\\n \\n <mrow>\\n <mo>(</mo>\\n \\n <mi>G</mi>\\n \\n <mo>)</mo>\\n </mrow>\\n \\n <mo>≤</mo>\\n \\n <mn>2</mn>\\n \\n <mi>Δ</mi>\\n </mrow>\\n </mrow>\\n <annotation> ${q}_{B}(G)\\\\le 2{\\\\rm{\\\\Delta }}$</annotation>\\n </semantics></math> for bipartite planar graphs, and <math>\\n <semantics>\\n <mrow>\\n \\n <mrow>\\n <msub>\\n <mi>q</mi>\\n \\n <mi>B</mi>\\n </msub>\\n \\n <mrow>\\n <mo>(</mo>\\n \\n <mi>G</mi>\\n \\n <mo>)</mo>\\n </mrow>\\n \\n <mo>≤</mo>\\n \\n <mi>Δ</mi>\\n \\n <mo>+</mo>\\n \\n <mn>1</mn>\\n </mrow>\\n </mrow>\\n <annotation> ${q}_{B}(G)\\\\le {\\\\rm{\\\\Delta }}+1$</annotation>\\n </semantics></math> for outerplanar graphs with <math>\\n <semantics>\\n <mrow>\\n \\n <mrow>\\n <mi>Δ</mi>\\n \\n <mo>≥</mo>\\n \\n <mn>4</mn>\\n </mrow>\\n </mrow>\\n <annotation> ${\\\\rm{\\\\Delta }}\\\\ge 4$</annotation>\\n </semantics></math>. We conjecture that, for <math>\\n <semantics>\\n <mrow>\\n \\n <mrow>\\n <mi>Δ</mi>\\n </mrow>\\n </mrow>\\n <annotation> ${\\\\rm{\\\\Delta }}$</annotation>\\n </semantics></math> sufficiently large, <math>\\n <semantics>\\n <mrow>\\n \\n <mrow>\\n <msub>\\n <mi>q</mi>\\n \\n <mi>B</mi>\\n </msub>\\n \\n <mrow>\\n <mo>(</mo>\\n \\n <mi>G</mi>\\n \\n <mo>)</mo>\\n </mrow>\\n \\n <mo>≤</mo>\\n \\n <mn>2</mn>\\n \\n <mi>Δ</mi>\\n \\n <mrow>\\n <mo>(</mo>\\n \\n <mi>G</mi>\\n \\n <mo>)</mo>\\n </mrow>\\n </mrow>\\n </mrow>\\n <annotation> ${q}_{B}(G)\\\\le 2{\\\\rm{\\\\Delta }}(G)$</annotation>\\n </semantics></math> for planar <math>\\n <semantics>\\n <mrow>\\n \\n <mrow>\\n <mi>G</mi>\\n </mrow>\\n </mrow>\\n <annotation> $G$</annotation>\\n </semantics></math>, and <math>\\n <semantics>\\n <mrow>\\n \\n <mrow>\\n <msub>\\n <mi>q</mi>\\n \\n <mi>B</mi>\\n </msub>\\n \\n <mrow>\\n <mo>(</mo>\\n \\n <mi>G</mi>\\n \\n <mo>)</mo>\\n </mrow>\\n \\n <mo>≤</mo>\\n \\n <mi>Δ</mi>\\n \\n <mrow>\\n <mo>(</mo>\\n \\n <mi>G</mi>\\n \\n <mo>)</mo>\\n </mrow>\\n </mrow>\\n </mrow>\\n <annotation> ${q}_{B}(G)\\\\le {\\\\rm{\\\\Delta }}(G)$</annotation>\\n </semantics></math> for outerplanar <math>\\n <semantics>\\n <mrow>\\n \\n <mrow>\\n <mi>G</mi>\\n </mrow>\\n </mrow>\\n <annotation> $G$</annotation>\\n </semantics></math>.\",\"PeriodicalId\":16014,\"journal\":{\"name\":\"Journal of Graph Theory\",\"volume\":\"107 4\",\"pages\":\"833-846\"},\"PeriodicalIF\":0.9000,\"publicationDate\":\"2024-08-05\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Journal of Graph Theory\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://onlinelibrary.wiley.com/doi/10.1002/jgt.23163\",\"RegionNum\":3,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Graph Theory","FirstCategoryId":"100","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1002/jgt.23163","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}

引用次数: 0

摘要

如果图的每个 4 循环都用四种不同的颜色着色，我们就称该图的适当边着色为 B 着色。让表示 B 染色所需的最小颜色数。受早先关于 B 染色的论文的启发，我们在此考虑平面图和外平面图的最大度。我们证明，对于平面图、双方形平面图以及具有 .我们猜想，在足够大的情况下，对于平面图 , 和外平面图 .

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Proper edge colorings of planar graphs with rainbow C 4 ${C}_{4}$ -s

We call a proper edge coloring of a graph $G$ a B-coloring if every 4-cycle of $G$ is colored with four different colors. Let $q_{B} (G)$ denote the smallest number of colors needed for a B-coloring of $G$ . Motivated by earlier papers on B-colorings, here we consider $q_{B} (G)$ for planar and outerplanar graphs in terms of the maximum degree $Δ = Δ (G)$ . We prove that $q_{B} (G) \leq 2 Δ + 8$ for planar graphs, $q_{B} (G) \leq 2 Δ$ for bipartite planar graphs, and $q_{B} (G) \leq Δ + 1$ for outerplanar graphs with $Δ \geq 4$ . We conjecture that, for $Δ$ sufficiently large, $q_{B} (G) \leq 2 Δ (G)$ for planar $G$ , and $q_{B} (G) \leq Δ (G)$ for outerplanar $G$ .

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来源期刊

Journal of Graph Theory 数学-数学

CiteScore

1.60

自引率

22.20%

发文量

130

审稿时长

6-12 weeks

期刊介绍： The Journal of Graph Theory is devoted to a variety of topics in graph theory, such as structural results about graphs, graph algorithms with theoretical emphasis, and discrete optimization on graphs. The scope of the journal also includes related areas in combinatorics and the interaction of graph theory with other mathematical sciences. A subscription to the Journal of Graph Theory includes a subscription to the Journal of Combinatorial Designs .