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.

中文翻译：

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完整图的彩虹饱和度

SIAM 离散数学杂志，第 38 卷，第 1 期，第 1090-1112 页，2024 年 3 月。
摘要。如果边缘彩色图的所有边缘都接收不同的颜色，我们将其称为彩虹。如果 [math] 不包含 [math] 的彩虹副本，并且向 [math] 添加任何颜色的边会创建 [math] 的彩虹副本，则边缘彩色图 [math] 称为 [math]-彩虹饱和。彩虹饱和数 [math] 是 [math]-顶点 [math]-彩虹饱和图中边的最小数量。Girão、Lewis 和 Popielarz 推测 [math] 为固定 [math]。反驳这个猜想，我们确定对于每个[数学]，都存在一个常数[数学]，使得[数学]和[数学]。最近，贝哈格、约翰斯顿、莱茨特、莫里森和奥格登独立给出了一个稍弱的上限，足以反驳这个猜想。他们还引入了弱彩虹饱和数，并询问这是否等于[数学]的彩虹饱和数，因为完整图的标准弱饱和数等于标准饱和数。令人惊讶的是，我们的下界将彩虹饱和度数与弱彩虹饱和度数分开，从而否定了这个问题。常数[数学]的存在解决了他们的另一个问题，对完全图是肯定的。此外，如果我们有一个额外的假设，即边缘颜色的[数学]-彩虹饱和图必须是彩虹，我们证明 Girão、Lewis 和 Popielarz 的猜想是正确的。作为证明的一个组成部分，我们研究在删除一条边和添加两条边的操作方面[数学]饱和的图。