{"title":"关于$$(k,g)$$ -图的降阶问题","authors":"Geoffrey Exoo, Robert Jajcay, Tom Raiman","doi":"10.1007/s10878-023-01092-9","DOIUrl":null,"url":null,"abstract":"<p>A <span>\\((k,g)\\)</span>-graph is a <i>k</i>-regular graph of girth <span>\\(g\\)</span>. Given <span>\\(k\\ge 2\\)</span> and <span>\\(g\\ge 3\\)</span>, <span>\\((k,g)\\)</span>-graphs of infinitely many orders are known to exist and the problem of finding a (<i>k</i>, <i>g</i>)-graph of the smallest possible order is known as the <i>Cage Problem</i>. The aim of our paper is to develop systematic (programmable) ways for lowering the orders of existing <span>\\((k,g)\\)</span>-graphs, while preserving their regularity and girth. Such methods, in analogy with the previously used excision, may have the potential for constructing smaller (<i>k</i>, <i>g</i>)-graphs from current smallest examples—record holders—some of which have not been improved in years. In addition, we consider constructions that preserve the regularity, the girth <i>and</i> the order of the considered graphs, but alter the graphs enough to possibly make them suitable for the application of our order decreasing methods. We include a detailed discussion of several specific parameter cases for which several non-isomorphic smallest examples are known to exist, and address the question of the distance between these non-isomorphic examples based on the number of changes required to move from one example to another.\n</p>","PeriodicalId":50231,"journal":{"name":"Journal of Combinatorial Optimization","volume":null,"pages":null},"PeriodicalIF":0.9000,"publicationDate":"2023-11-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"On decreasing the orders of $$(k,g)$$ -graphs\",\"authors\":\"Geoffrey Exoo, Robert Jajcay, Tom Raiman\",\"doi\":\"10.1007/s10878-023-01092-9\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>A <span>\\\\((k,g)\\\\)</span>-graph is a <i>k</i>-regular graph of girth <span>\\\\(g\\\\)</span>. Given <span>\\\\(k\\\\ge 2\\\\)</span> and <span>\\\\(g\\\\ge 3\\\\)</span>, <span>\\\\((k,g)\\\\)</span>-graphs of infinitely many orders are known to exist and the problem of finding a (<i>k</i>, <i>g</i>)-graph of the smallest possible order is known as the <i>Cage Problem</i>. The aim of our paper is to develop systematic (programmable) ways for lowering the orders of existing <span>\\\\((k,g)\\\\)</span>-graphs, while preserving their regularity and girth. Such methods, in analogy with the previously used excision, may have the potential for constructing smaller (<i>k</i>, <i>g</i>)-graphs from current smallest examples—record holders—some of which have not been improved in years. In addition, we consider constructions that preserve the regularity, the girth <i>and</i> the order of the considered graphs, but alter the graphs enough to possibly make them suitable for the application of our order decreasing methods. We include a detailed discussion of several specific parameter cases for which several non-isomorphic smallest examples are known to exist, and address the question of the distance between these non-isomorphic examples based on the number of changes required to move from one example to another.\\n</p>\",\"PeriodicalId\":50231,\"journal\":{\"name\":\"Journal of Combinatorial Optimization\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.9000,\"publicationDate\":\"2023-11-16\",\"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/s10878-023-01092-9\",\"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}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Combinatorial Optimization","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s10878-023-01092-9","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}
A \((k,g)\)-graph is a k-regular graph of girth \(g\). Given \(k\ge 2\) and \(g\ge 3\), \((k,g)\)-graphs of infinitely many orders are known to exist and the problem of finding a (k, g)-graph of the smallest possible order is known as the Cage Problem. The aim of our paper is to develop systematic (programmable) ways for lowering the orders of existing \((k,g)\)-graphs, while preserving their regularity and girth. Such methods, in analogy with the previously used excision, may have the potential for constructing smaller (k, g)-graphs from current smallest examples—record holders—some of which have not been improved in years. In addition, we consider constructions that preserve the regularity, the girth and the order of the considered graphs, but alter the graphs enough to possibly make them suitable for the application of our order decreasing methods. We include a detailed discussion of several specific parameter cases for which several non-isomorphic smallest examples are known to exist, and address the question of the distance between these non-isomorphic examples based on the number of changes required to move from one example to another.
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.