Debsoumya Chakraborti, Kevin Hendrey, Ben Lund, Casey Tompkins
{"title":"完整图形的彩虹饱和度","authors":"Debsoumya Chakraborti, Kevin Hendrey, Ben Lund, Casey Tompkins","doi":"10.1137/23m1565875","DOIUrl":null,"url":null,"abstract":"SIAM Journal on Discrete Mathematics, Volume 38, Issue 1, Page 1090-1112, March 2024. <br/> Abstract. We call an edge-colored graph rainbow if all of its edges receive distinct colors. An edge-colored graph [math] is called [math]-rainbow saturated if [math] does not contain a rainbow copy of [math] and adding an edge of any color to [math] creates a rainbow copy of [math]. The rainbow saturation number [math] is the minimum number of edges in an [math]-vertex [math]-rainbow saturated graph. Girão, Lewis, and Popielarz conjectured that [math] for fixed [math]. Disproving this conjecture, we establish that for every [math], there exists a constant [math] such that [math] and [math]. Recently, Behague, Johnston, Letzter, Morrison, and Ogden independently gave a slightly weaker upper bound which was sufficient to disprove the conjecture. They also introduced the weak rainbow saturation number and asked whether this is equal to the rainbow saturation number of [math], since the standard weak saturation number of complete graphs equals the standard saturation number. Surprisingly, our lower bound separates the rainbow saturation number from the weak rainbow saturation number, answering this question in the negative. The existence of the constant [math] resolves another of their questions in the affirmative for complete graphs. Furthermore, we show that the conjecture of Girão, Lewis, and Popielarz is true if we have an additional assumption that the edge-colored [math]-rainbow saturated graph must be rainbow. As an ingredient of the proof, we study graphs which are [math]-saturated with respect to the operation of deleting one edge and adding two edges.","PeriodicalId":49530,"journal":{"name":"SIAM Journal on Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.9000,"publicationDate":"2024-03-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Rainbow Saturation for Complete Graphs\",\"authors\":\"Debsoumya Chakraborti, Kevin Hendrey, Ben Lund, Casey Tompkins\",\"doi\":\"10.1137/23m1565875\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"SIAM Journal on Discrete Mathematics, Volume 38, Issue 1, Page 1090-1112, March 2024. <br/> Abstract. We call an edge-colored graph rainbow if all of its edges receive distinct colors. An edge-colored graph [math] is called [math]-rainbow saturated if [math] does not contain a rainbow copy of [math] and adding an edge of any color to [math] creates a rainbow copy of [math]. The rainbow saturation number [math] is the minimum number of edges in an [math]-vertex [math]-rainbow saturated graph. Girão, Lewis, and Popielarz conjectured that [math] for fixed [math]. Disproving this conjecture, we establish that for every [math], there exists a constant [math] such that [math] and [math]. Recently, Behague, Johnston, Letzter, Morrison, and Ogden independently gave a slightly weaker upper bound which was sufficient to disprove the conjecture. They also introduced the weak rainbow saturation number and asked whether this is equal to the rainbow saturation number of [math], since the standard weak saturation number of complete graphs equals the standard saturation number. Surprisingly, our lower bound separates the rainbow saturation number from the weak rainbow saturation number, answering this question in the negative. The existence of the constant [math] resolves another of their questions in the affirmative for complete graphs. Furthermore, we show that the conjecture of Girão, Lewis, and Popielarz is true if we have an additional assumption that the edge-colored [math]-rainbow saturated graph must be rainbow. As an ingredient of the proof, we study graphs which are [math]-saturated with respect to the operation of deleting one edge and adding two edges.\",\"PeriodicalId\":49530,\"journal\":{\"name\":\"SIAM Journal on Discrete Mathematics\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.9000,\"publicationDate\":\"2024-03-14\",\"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/10.1137/23m1565875\",\"RegionNum\":3,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"SIAM Journal on Discrete Mathematics","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1137/23m1565875","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
SIAM Journal on Discrete Mathematics, Volume 38, Issue 1, Page 1090-1112, March 2024. Abstract. We call an edge-colored graph rainbow if all of its edges receive distinct colors. An edge-colored graph [math] is called [math]-rainbow saturated if [math] does not contain a rainbow copy of [math] and adding an edge of any color to [math] creates a rainbow copy of [math]. The rainbow saturation number [math] is the minimum number of edges in an [math]-vertex [math]-rainbow saturated graph. Girão, Lewis, and Popielarz conjectured that [math] for fixed [math]. Disproving this conjecture, we establish that for every [math], there exists a constant [math] such that [math] and [math]. Recently, Behague, Johnston, Letzter, Morrison, and Ogden independently gave a slightly weaker upper bound which was sufficient to disprove the conjecture. They also introduced the weak rainbow saturation number and asked whether this is equal to the rainbow saturation number of [math], since the standard weak saturation number of complete graphs equals the standard saturation number. Surprisingly, our lower bound separates the rainbow saturation number from the weak rainbow saturation number, answering this question in the negative. The existence of the constant [math] resolves another of their questions in the affirmative for complete graphs. Furthermore, we show that the conjecture of Girão, Lewis, and Popielarz is true if we have an additional assumption that the edge-colored [math]-rainbow saturated graph must be rainbow. As an ingredient of the proof, we study graphs which are [math]-saturated with respect to the operation of deleting one edge and adding two edges.
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.