{"title":"On the metric subgraphs of a graph","authors":"Yanan Hu, Xingzhi Zhan","doi":"10.26493/1855-3974.2992.a6d","DOIUrl":null,"url":null,"abstract":"The three subgraphs of a connected graph induced by the center, annulus and periphery are called its metric subgraphs. The main results are as follows. (1) There exists a graph of order n whose metric subgraphs are all paths if and only if n ≥ 13 and the smallest size of such a graph of order 13 is 22; (2) there exists a graph of order n whose metric subgraphs are all cycles if and only if n ≥ 15, and there are exactly three such graphs of order 15; (3) for every integer k ≥ 3, we determine the possible orders for the existence of a graph whose metric subgraphs are all connected k-regular graphs; (4) there exists a graph of order n whose metric subgraphs are connected and pairwise isomorphic if and only if n ≥ 24 and n is divisible by 3. An unsolved problem is posed.","PeriodicalId":49239,"journal":{"name":"Ars Mathematica Contemporanea","volume":"135 3","pages":"0"},"PeriodicalIF":0.6000,"publicationDate":"2023-11-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Ars Mathematica Contemporanea","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.26493/1855-3974.2992.a6d","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
The three subgraphs of a connected graph induced by the center, annulus and periphery are called its metric subgraphs. The main results are as follows. (1) There exists a graph of order n whose metric subgraphs are all paths if and only if n ≥ 13 and the smallest size of such a graph of order 13 is 22; (2) there exists a graph of order n whose metric subgraphs are all cycles if and only if n ≥ 15, and there are exactly three such graphs of order 15; (3) for every integer k ≥ 3, we determine the possible orders for the existence of a graph whose metric subgraphs are all connected k-regular graphs; (4) there exists a graph of order n whose metric subgraphs are connected and pairwise isomorphic if and only if n ≥ 24 and n is divisible by 3. An unsolved problem is posed.
期刊介绍:
Ars mathematica contemporanea will publish high-quality articles in contemporary mathematics that arise from the discrete and concrete mathematics paradigm. It will favor themes that combine at least two different fields of mathematics. In particular, we welcome papers intersecting discrete mathematics with other branches of mathematics, such as algebra, geometry, topology, theoretical computer science, and combinatorics. The name of the journal was chosen carefully. Symmetry is certainly a theme that is quite welcome to the journal, as it is through symmetry that mathematics comes closest to art.