{"title":"A tight upper bound on the average order of dominating sets of a graph","authors":"Iain Beaton, Ben Cameron","doi":"10.1002/jgt.23143","DOIUrl":null,"url":null,"abstract":"<p>In this paper we study the average order of dominating sets in a graph, <span></span><math>\n <semantics>\n <mrow>\n \n <mrow>\n <mstyle>\n <mspace></mspace>\n \n <mtext>avd</mtext>\n <mspace></mspace>\n </mstyle>\n \n <mrow>\n <mo>(</mo>\n \n <mi>G</mi>\n \n <mo>)</mo>\n </mrow>\n </mrow>\n </mrow>\n <annotation> $\\,\\text{avd}\\,(G)$</annotation>\n </semantics></math>. Like other average graph parameters, the extremal graphs are of interest. Beaton and Brown conjectured that for all graphs <span></span><math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>G</mi>\n </mrow>\n </mrow>\n <annotation> $G$</annotation>\n </semantics></math> of order <span></span><math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>n</mi>\n </mrow>\n </mrow>\n <annotation> $n$</annotation>\n </semantics></math> without isolated vertices, <span></span><math>\n <semantics>\n <mrow>\n \n <mrow>\n <mspace></mspace>\n \n <mtext>avd</mtext>\n <mspace></mspace>\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>n</mi>\n \n <mo>/</mo>\n \n <mn>3</mn>\n </mrow>\n </mrow>\n <annotation> $\\,\\text{avd}\\,(G)\\le 2n/3$</annotation>\n </semantics></math>. Recently, Erey proved the conjecture for forests without isolated vertices. In this paper we prove the conjecture and classify which graphs have <span></span><math>\n <semantics>\n <mrow>\n \n <mrow>\n <mspace></mspace>\n \n <mtext>avd</mtext>\n <mspace></mspace>\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>n</mi>\n \n <mo>/</mo>\n \n <mn>3</mn>\n </mrow>\n </mrow>\n <annotation> $\\,\\text{avd}\\,(G)=2n/3$</annotation>\n </semantics></math>. We also use our bounds to prove an average version of Vizing's conjecture.</p>","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"107 3","pages":"463-477"},"PeriodicalIF":0.9000,"publicationDate":"2024-06-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/jgt.23143","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Graph Theory","FirstCategoryId":"100","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1002/jgt.23143","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
In this paper we study the average order of dominating sets in a graph, . Like other average graph parameters, the extremal graphs are of interest. Beaton and Brown conjectured that for all graphs of order without isolated vertices, . Recently, Erey proved the conjecture for forests without isolated vertices. In this paper we prove the conjecture and classify which graphs have . We also use our bounds to prove an average version of Vizing's conjecture.
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 .