{"title":"The minimum orientable genus of the repeated Cartesian product of graphs","authors":"Marietta Galea, John Baptist Gauci","doi":"10.1007/s10878-025-01266-7","DOIUrl":null,"url":null,"abstract":"<p>Determining the minimum genus of a graph is a fundamental optimisation problem in the study of network design and implementation as it gives a measure of non-planarity of graphs. In this paper, we are concerned with determining the smallest value of <i>g</i> such that a given graph <i>G</i> has an embedding on the orientable surface of genus <i>g</i>. In particular, we consider the Cartesian product of graphs since this is a well studied graph operation which is often used for modeling interconnection networks. The <i>s</i>-cube <span>\\(Q_i^{(s)}\\)</span> is obtained by taking the repeated Cartesian product of <i>i</i> complete bipartite graphs <span>\\(K_{s,s}\\)</span>. We determine the genus of the Cartesian product of the 2<i>r</i>-cube with the repeated Cartesian product of cycles and of the Cartesian product of the 2<i>r</i>-cube with the repeated Cartesian product of paths.</p>","PeriodicalId":50231,"journal":{"name":"Journal of Combinatorial Optimization","volume":"81 1","pages":""},"PeriodicalIF":0.9000,"publicationDate":"2025-02-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Combinatorial Optimization","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s10878-025-01266-7","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS","Score":null,"Total":0}
引用次数: 0
Abstract
Determining the minimum genus of a graph is a fundamental optimisation problem in the study of network design and implementation as it gives a measure of non-planarity of graphs. In this paper, we are concerned with determining the smallest value of g such that a given graph G has an embedding on the orientable surface of genus g. In particular, we consider the Cartesian product of graphs since this is a well studied graph operation which is often used for modeling interconnection networks. The s-cube \(Q_i^{(s)}\) is obtained by taking the repeated Cartesian product of i complete bipartite graphs \(K_{s,s}\). We determine the genus of the Cartesian product of the 2r-cube with the repeated Cartesian product of cycles and of the Cartesian product of the 2r-cube with the repeated Cartesian product of paths.
期刊介绍:
The objective of Journal of Combinatorial Optimization is to advance and promote the theory and applications of combinatorial optimization, which is an area of research at the intersection of applied mathematics, computer science, and operations research and which overlaps with many other areas such as computation complexity, computational biology, VLSI design, communication networks, and management science. It includes complexity analysis and algorithm design for combinatorial optimization problems, numerical experiments and problem discovery with applications in science and engineering.
The Journal of Combinatorial Optimization publishes refereed papers dealing with all theoretical, computational and applied aspects of combinatorial optimization. It also publishes reviews of appropriate books and special issues of journals.