A tight upper bound on the average order of dominating sets of a graph

Pub Date : 2024-06-17 DOI:10.1002/jgt.23143

Iain Beaton, Ben Cameron

{"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":"In this paper we study the average order of dominating sets in a graph, <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 <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 <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, <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 <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.","PeriodicalId":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"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":"","FirstCategoryId":"100","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1002/jgt.23143","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}

引用次数: 0

Abstract

In this paper we study the average order of dominating sets in a graph, $avd (G)$ . Like other average graph parameters, the extremal graphs are of interest. Beaton and Brown conjectured that for all graphs $G$ of order $n$ without isolated vertices, $avd (G) \leq 2 n / 3$ . Recently, Erey proved the conjecture for forests without isolated vertices. In this paper we prove the conjecture and classify which graphs have $avd (G) = 2 n / 3$ . We also use our bounds to prove an average version of Vizing's conjecture.

Abstract Image

查看原文

图的支配集平均阶数的严格上限

本文研究图中支配集的平均阶数 avd ( G ) $\,\text{avd}\,(G)$ 。与其他平均图参数一样，极值图也很值得关注。比顿和布朗猜想，对于所有阶数为 n $n$ 的无孤立顶点的图 G $G$ ，avd ( G ) ≤ 2 n / 3 $,\text{avd}\,(G)\le 2n/3$ 。最近，埃雷证明了无孤立顶点森林的猜想。在本文中，我们证明了这个猜想，并分类了哪些图具有 avd ( G ) = 2 n / 3 $\,\text{avd}\,(G)=2n/3$ 。我们还利用我们的边界证明了平均版本的 Vizing 猜想。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

求助全文

约1分钟内获得全文求助全文