{"title":"Counting circuit double covers","authors":"Radek Hušek, Robert Šámal","doi":"10.1002/jgt.23187","DOIUrl":null,"url":null,"abstract":"<p>We study a counting version of Cycle Double Cover Conjecture. We discuss why it is more interesting to count circuits (i.e., graphs isomorphic to <span></span><math>\n <semantics>\n <mrow>\n \n <mrow>\n <msub>\n <mi>C</mi>\n \n <mi>k</mi>\n </msub>\n </mrow>\n </mrow>\n <annotation> ${C}_{k}$</annotation>\n </semantics></math> for some <span></span><math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>k</mi>\n </mrow>\n </mrow>\n <annotation> $k$</annotation>\n </semantics></math>) instead of cycles (graphs with all degrees even). We give an almost-exponential lower bound for graphs with a surface embedding of representativity at least 4. We also prove an exponential lower bound for planar graphs. We conjecture that any bridgeless cubic graph has at least <span></span><math>\n <semantics>\n <mrow>\n \n <mrow>\n <msup>\n <mn>2</mn>\n \n <mrow>\n <mi>n</mi>\n \n <mo>/</mo>\n \n <mn>2</mn>\n \n <mo>−</mo>\n \n <mn>1</mn>\n </mrow>\n </msup>\n </mrow>\n </mrow>\n <annotation> ${2}^{n/2-1}$</annotation>\n </semantics></math> circuit double covers and we show an infinite class of graphs for which this bound is tight.</p>","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"108 2","pages":"374-395"},"PeriodicalIF":0.9000,"publicationDate":"2024-10-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/jgt.23187","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Graph Theory","FirstCategoryId":"100","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1002/jgt.23187","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
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Abstract
We study a counting version of Cycle Double Cover Conjecture. We discuss why it is more interesting to count circuits (i.e., graphs isomorphic to for some ) instead of cycles (graphs with all degrees even). We give an almost-exponential lower bound for graphs with a surface embedding of representativity at least 4. We also prove an exponential lower bound for planar graphs. We conjecture that any bridgeless cubic graph has at least circuit double covers and we show an infinite class of graphs for which this bound is tight.
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 .
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