{"title":"Phase Transitions of Structured Codes of Graphs","authors":"Bo Bai, Yu Gao, Jie Ma, Yuze Wu","doi":"10.1137/23m1614572","DOIUrl":null,"url":null,"abstract":"SIAM Journal on Discrete Mathematics, Volume 38, Issue 2, Page 1902-1914, June 2024. <br/> Abstract. We consider the symmetric difference of two graphs on the same vertex set [math], which is the graph on [math] whose edge set consists of all edges that belong to exactly one of the two graphs. Let [math] be a class of graphs, and let [math] denote the maximum possible cardinality of a family [math] of graphs on [math] such that the symmetric difference of any two members in [math] belongs to [math]. These concepts have been recently investigated by Alon et al. [SIAM J. Discrete Math., 37 (2023), pp. 379–403] with the aim of providing a new graphic approach to coding theory. In particular, [math] denotes the maximum possible size of this code. Existing results show that as the graph class [math] changes, [math] can vary from [math] to [math]. We study several phase transition problems related to [math] in general settings and present a partial solution to a recent problem posed by Alon et al.","PeriodicalId":49530,"journal":{"name":"SIAM Journal on Discrete Mathematics","volume":"27 1","pages":""},"PeriodicalIF":1.0000,"publicationDate":"2024-06-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"SIAM Journal on Discrete Mathematics","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1137/23m1614572","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
SIAM Journal on Discrete Mathematics, Volume 38, Issue 2, Page 1902-1914, June 2024. Abstract. We consider the symmetric difference of two graphs on the same vertex set [math], which is the graph on [math] whose edge set consists of all edges that belong to exactly one of the two graphs. Let [math] be a class of graphs, and let [math] denote the maximum possible cardinality of a family [math] of graphs on [math] such that the symmetric difference of any two members in [math] belongs to [math]. These concepts have been recently investigated by Alon et al. [SIAM J. Discrete Math., 37 (2023), pp. 379–403] with the aim of providing a new graphic approach to coding theory. In particular, [math] denotes the maximum possible size of this code. Existing results show that as the graph class [math] changes, [math] can vary from [math] to [math]. We study several phase transition problems related to [math] in general settings and present a partial solution to a recent problem posed by Alon et al.
期刊介绍：
SIAM Journal on Discrete Mathematics (SIDMA) publishes research papers of exceptional quality in pure and applied discrete mathematics, broadly interpreted. The journal''s focus is primarily theoretical rather than empirical, but the editors welcome papers that evolve from or have potential application to real-world problems. Submissions must be clearly written and make a significant contribution.
Topics include but are not limited to:
properties of and extremal problems for discrete structures
combinatorial optimization, including approximation algorithms
algebraic and enumerative combinatorics
coding and information theory
additive, analytic combinatorics and number theory
combinatorial matrix theory and spectral graph theory
design and analysis of algorithms for discrete structures
discrete problems in computational complexity
discrete and computational geometry
discrete methods in computational biology, and bioinformatics
probabilistic methods and randomized algorithms.
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