The Rainbow Saturation Number Is Linear

IF 0.9 3区数学 Q2 MATHEMATICS

SIAM Journal on Discrete Mathematics Pub Date : 2024-04-08 DOI:10.1137/23m1566881

Natalie Behague, Tom Johnston, Shoham Letzter, Natasha Morrison, Shannon Ogden

引用次数: 0

Abstract

SIAM Journal on Discrete Mathematics, Volume 38, Issue 2, Page 1239-1249, June 2024.
Abstract. Given a graph [math], we say that an edge-colored graph [math] is [math]-rainbow saturated if it does not contain a rainbow copy of [math], but the addition of any nonedge in any color creates a rainbow copy of [math]. The rainbow saturation number [math] is the minimum number of edges among all [math]-rainbow saturated edge-colored graphs on [math] vertices. We prove that for any nonempty graph [math], the rainbow saturation number is linear in [math], thus proving a conjecture of Girão, Lewis, and Popielarz. In addition, we give an improved upper bound on the rainbow saturation number of the complete graph, disproving a second conjecture of Girão, Lewis, and Popielarz.

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彩虹饱和度数值是线性的

SIAM 离散数学杂志》第 38 卷第 2 期第 1239-1249 页，2024 年 6 月。摘要。给定一个图[math]，如果一个边色图[math]不包含[math]的彩虹副本，但添加任何颜色的非边都会产生[math]的彩虹副本，我们就说这个边色图[math]是[math]-彩虹饱和的。彩虹饱和数[math]是[math]顶点上所有[math]-彩虹饱和边色图中最少的边数。我们证明了对于任何非空图 [math]，彩虹饱和度数在 [math] 中是线性的，从而证明了 Girão、Lewis 和 Popielarz 的猜想。此外，我们还给出了完整图的彩虹饱和数的改进上界，推翻了吉朗、刘易斯和波皮拉尔兹的第二个猜想。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

SIAM Journal on Discrete Mathematics 数学-应用数学

CiteScore

1.90

自引率

0.00%

发文量

124

审稿时长

4-8 weeks

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.