Weak Rainbow Saturation Numbers of Graphs

IF 0.9 3区数学 Q2 MATHEMATICS

Journal of Graph Theory Pub Date : 2025-01-06 DOI:10.1002/jgt.23211

Xihe Li, Jie Ma, Tianying Xie

{"title":"Weak Rainbow Saturation Numbers of Graphs","authors":"Xihe Li, Jie Ma, Tianying Xie","doi":"10.1002/jgt.23211","DOIUrl":null,"url":null,"abstract":"<div>\n \n For a fixed graph <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>H</mi>\n </mrow>\n </mrow>\n <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt23211:jgt23211-math-0001\" wiley:location=\"equation/jgt23211-math-0001.png\"><mrow><mrow><mi>H</mi></mrow></mrow></math></annotation>\n </semantics></math>, we say that an edge-colored graph <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>G</mi>\n </mrow>\n </mrow>\n <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt23211:jgt23211-math-0002\" wiley:location=\"equation/jgt23211-math-0002.png\"><mrow><mrow><mi>G</mi></mrow></mrow></math></annotation>\n </semantics></math> is weakly <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>H</mi>\n </mrow>\n </mrow>\n <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt23211:jgt23211-math-0003\" wiley:location=\"equation/jgt23211-math-0003.png\"><mrow><mrow><mi>H</mi></mrow></mrow></math></annotation>\n </semantics></math>-rainbow saturated if there exists an ordering <math>\n <semantics>\n <mrow>\n \n <mrow>\n <msub>\n <mi>e</mi>\n \n <mn>1</mn>\n </msub>\n \n <mo>,</mo>\n \n <msub>\n <mi>e</mi>\n \n <mn>2</mn>\n </msub>\n \n <mo>,</mo>\n \n <mo>…</mo>\n \n <mo>,</mo>\n \n <msub>\n <mi>e</mi>\n \n <mi>m</mi>\n </msub>\n </mrow>\n </mrow>\n <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt23211:jgt23211-math-0004\" wiley:location=\"equation/jgt23211-math-0004.png\"><mrow><mrow><msub><mi>e</mi><mn>1</mn></msub><mo>,</mo><msub><mi>e</mi><mn>2</mn></msub><mo>,</mo><mo>\\unicode{x02026}</mo><mo>,</mo><msub><mi>e</mi><mi>m</mi></msub></mrow></mrow></math></annotation>\n </semantics></math> of <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>E</mi>\n \n <mrow>\n <mo>(</mo>\n \n <mover>\n <mi>G</mi>\n \n <mo>¯</mo>\n </mover>\n \n <mo>)</mo>\n </mrow>\n </mrow>\n </mrow>\n <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt23211:jgt23211-math-0005\" wiley:location=\"equation/jgt23211-math-0005.png\"><mrow><mrow><mi>E</mi><mrow><mo stretchy=\"false\">(</mo><mover accent=\"true\"><mi>G</mi><mo stretchy=\"true\">\\unicode{x000AF}</mo></mover><mo stretchy=\"false\">)</mo></mrow></mrow></mrow></math></annotation>\n </semantics></math> such that, for any list <math>\n <semantics>\n <mrow>\n \n <mrow>\n <msub>\n <mi>c</mi>\n \n <mn>1</mn>\n </msub>\n \n <mo>,</mo>\n \n <msub>\n <mi>c</mi>\n \n <mn>2</mn>\n </msub>\n \n <mo>,</mo>\n \n <mo>…</mo>\n \n <mo>,</mo>\n \n <msub>\n <mi>c</mi>\n \n <mi>m</mi>\n </msub>\n </mrow>\n </mrow>\n <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt23211:jgt23211-math-0006\" wiley:location=\"equation/jgt23211-math-0006.png\"><mrow><mrow><msub><mi>c</mi><mn>1</mn></msub><mo>,</mo><msub><mi>c</mi><mn>2</mn></msub><mo>,</mo><mo>\\unicode{x02026}</mo><mo>,</mo><msub><mi>c</mi><mi>m</mi></msub></mrow></mrow></math></annotation>\n </semantics></math> of pairwise distinct colors from <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>N</mi>\n </mrow>\n </mrow>\n <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt23211:jgt23211-math-0007\" wiley:location=\"equation/jgt23211-math-0007.png\"><mrow><mrow><mi mathvariant=\"double-struck\">N</mi></mrow></mrow></math></annotation>\n </semantics></math>, the nonedges <math>\n <semantics>\n <mrow>\n \n <mrow>\n <msub>\n <mi>e</mi>\n \n <mi>i</mi>\n </msub>\n </mrow>\n </mrow>\n <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt23211:jgt23211-math-0008\" wiley:location=\"equation/jgt23211-math-0008.png\"><mrow><mrow><msub><mi>e</mi><mi>i</mi></msub></mrow></mrow></math></annotation>\n </semantics></math> in color <math>\n <semantics>\n <mrow>\n \n <mrow>\n <msub>\n <mi>c</mi>\n \n <mi>i</mi>\n </msub>\n </mrow>\n </mrow>\n <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt23211:jgt23211-math-0009\" wiley:location=\"equation/jgt23211-math-0009.png\"><mrow><mrow><msub><mi>c</mi><mi>i</mi></msub></mrow></mrow></math></annotation>\n </semantics></math> can be added to <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>G</mi>\n </mrow>\n </mrow>\n <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt23211:jgt23211-math-0010\" wiley:location=\"equation/jgt23211-math-0010.png\"><mrow><mrow><mi>G</mi></mrow></mrow></math></annotation>\n </semantics></math>, one at a time, so that every added edge creates a new rainbow copy of <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>H</mi>\n </mrow>\n </mrow>\n <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt23211:jgt23211-math-0011\" wiley:location=\"equation/jgt23211-math-0011.png\"><mrow><mrow><mi>H</mi></mrow></mrow></math></annotation>\n </semantics></math>. The weak rainbow saturation number of <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>H</mi>\n </mrow>\n </mrow>\n <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt23211:jgt23211-math-0012\" wiley:location=\"equation/jgt23211-math-0012.png\"><mrow><mrow><mi>H</mi></mrow></mrow></math></annotation>\n </semantics></math>, denoted by <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mtext>rwsat</mtext>\n \n <mrow>\n <mo>(</mo>\n \n <mrow>\n <mi>n</mi>\n \n <mo>,</mo>\n \n <mi>H</mi>\n </mrow>\n \n <mo>)</mo>\n </mrow>\n </mrow>\n </mrow>\n <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt23211:jgt23211-math-0013\" wiley:location=\"equation/jgt23211-math-0013.png\"><mrow><mrow><mtext>rwsat</mtext><mrow><mo>(</mo><mrow><mi>n</mi><mo>,</mo><mi>H</mi></mrow><mo>)</mo></mrow></mrow></mrow></math></annotation>\n </semantics></math>, is the minimum number of edges in a weakly <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>H</mi>\n </mrow>\n </mrow>\n <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt23211:jgt23211-math-0014\" wiley:location=\"equation/jgt23211-math-0014.png\"><mrow><mrow><mi>H</mi></mrow></mrow></math></annotation>\n </semantics></math>-rainbow saturated graph on <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>n</mi>\n </mrow>\n </mrow>\n <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt23211:jgt23211-math-0015\" wiley:location=\"equation/jgt23211-math-0015.png\"><mrow><mrow><mi>n</mi></mrow></mrow></math></annotation>\n </semantics></math> vertices. In this paper, we show that for any nonempty graph <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>H</mi>\n </mrow>\n </mrow>\n <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt23211:jgt23211-math-0016\" wiley:location=\"equation/jgt23211-math-0016.png\"><mrow><mrow><mi>H</mi></mrow></mrow></math></annotation>\n </semantics></math>, the limit <math>\n <semantics>\n <mrow>\n \n <mrow>\n <msub>\n <mi>lim</mi>\n \n <mrow>\n <mi>n</mi>\n \n <mo>→</mo>\n \n <mi>∞</mi>\n </mrow>\n </msub>\n \n <mfrac>\n <mrow>\n <mtext>rwsat</mtext>\n \n <mrow>\n <mo>(</mo>\n \n <mrow>\n <mi>n</mi>\n \n <mo>,</mo>\n \n <mi>H</mi>\n </mrow>\n \n <mo>)</mo>\n </mrow>\n </mrow>\n \n <mi>n</mi>\n </mfrac>\n </mrow>\n </mrow>\n <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt23211:jgt23211-math-0017\" wiley:location=\"equation/jgt23211-math-0017.png\"><mrow><mrow><msub><mi>lim</mi><mrow><mi>n</mi><mo>\\unicode{x02192}</mo><mi>\\unicode{x0221E}</mi></mrow></msub><mfrac><mrow><mtext>rwsat</mtext><mrow><mo>(</mo><mrow><mi>n</mi><mo>,</mo><mi>H</mi></mrow><mo>)</mo></mrow></mrow><mi>n</mi></mfrac></mrow></mrow></math></annotation>\n </semantics></math> exists. This answers a question of Behague et al. We also provide lower and upper bounds on this limit, and in particular, we show that this limit is nonzero if and only if <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>H</mi>\n </mrow>\n </mrow>\n <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt23211:jgt23211-math-0018\" wiley:location=\"equation/jgt23211-math-0018.png\"><mrow><mrow><mi>H</mi></mrow></mrow></math></annotation>\n </semantics></math> contains no pendant edges.\n </div>","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"109 1","pages":"35-42"},"PeriodicalIF":0.9000,"publicationDate":"2025-01-06","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.23211","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

Abstract

For a fixed graph $H$ , we say that an edge-colored graph $G$ is weakly $H$ -rainbow saturated if there exists an ordering $e_{1}, e_{2}, \dots, e_{m}$ of $E (\bar{G})$ such that, for any list $c_{1}, c_{2}, \dots, c_{m}$ of pairwise distinct colors from $N$ , the nonedges $e_{i}$ in color $c_{i}$ can be added to $G$ , one at a time, so that every added edge creates a new rainbow copy of $H$ . The weak rainbow saturation number of $H$ , denoted by $rwsat (n, H)$ , is the minimum number of edges in a weakly $H$ -rainbow saturated graph on $n$ vertices. In this paper, we show that for any nonempty graph $H$ , the limit $\lim_{n \to \infty} \frac{rwsat (n, H)}{n}$ exists. This answers a question of Behague et al. We also provide lower and upper bounds on this limit, and in particular, we show that this limit is nonzero if and only if $H$ contains no pendant edges.

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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 .