N. Alon, Anna Gujgiczer, J. Körner, Aleksa Milojević, G. Simonyi
{"title":"图的结构化代码","authors":"N. Alon, Anna Gujgiczer, J. Körner, Aleksa Milojević, G. Simonyi","doi":"10.1137/22m1487989","DOIUrl":null,"url":null,"abstract":"We investigate the maximum size of graph families on a common vertex set of cardinality $n$ such that the symmetric difference of the edge sets of any two members of the family satisfies some prescribed condition. We solve the problem completely for infinitely many values of $n$ when the prescribed condition is connectivity or $2$-connectivity, Hamiltonicity or the containment of a spanning star. We also investigate local conditions that can be certified by looking at only a subset of the vertex set. In these cases a capacity-type asymptotic invariant is defined and when the condition is to contain a certain subgraph this invariant is shown to be a simple function of the chromatic number of this required subgraph. This is proven using classical results from extremal graph theory. Several variants are considered and the paper ends with a collection of open problems.","PeriodicalId":21749,"journal":{"name":"SIAM J. Discret. Math.","volume":"91 1","pages":"379-403"},"PeriodicalIF":0.0000,"publicationDate":"2022-02-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"6","resultStr":"{\"title\":\"Structured Codes of Graphs\",\"authors\":\"N. Alon, Anna Gujgiczer, J. Körner, Aleksa Milojević, G. Simonyi\",\"doi\":\"10.1137/22m1487989\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"We investigate the maximum size of graph families on a common vertex set of cardinality $n$ such that the symmetric difference of the edge sets of any two members of the family satisfies some prescribed condition. We solve the problem completely for infinitely many values of $n$ when the prescribed condition is connectivity or $2$-connectivity, Hamiltonicity or the containment of a spanning star. We also investigate local conditions that can be certified by looking at only a subset of the vertex set. In these cases a capacity-type asymptotic invariant is defined and when the condition is to contain a certain subgraph this invariant is shown to be a simple function of the chromatic number of this required subgraph. This is proven using classical results from extremal graph theory. Several variants are considered and the paper ends with a collection of open problems.\",\"PeriodicalId\":21749,\"journal\":{\"name\":\"SIAM J. Discret. Math.\",\"volume\":\"91 1\",\"pages\":\"379-403\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2022-02-14\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"6\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"SIAM J. Discret. Math.\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.1137/22m1487989\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"SIAM J. Discret. Math.","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1137/22m1487989","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
We investigate the maximum size of graph families on a common vertex set of cardinality $n$ such that the symmetric difference of the edge sets of any two members of the family satisfies some prescribed condition. We solve the problem completely for infinitely many values of $n$ when the prescribed condition is connectivity or $2$-connectivity, Hamiltonicity or the containment of a spanning star. We also investigate local conditions that can be certified by looking at only a subset of the vertex set. In these cases a capacity-type asymptotic invariant is defined and when the condition is to contain a certain subgraph this invariant is shown to be a simple function of the chromatic number of this required subgraph. This is proven using classical results from extremal graph theory. Several variants are considered and the paper ends with a collection of open problems.