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Rainbow saturation of graphs
Journal of Graph Theory ( IF 0.9 ) Pub Date : 2019-12-20 , DOI: 10.1002/jgt.22532
António Girão 1 , David Lewis 2 , Kamil Popielarz 2
Affiliation  

In this paper we study the following problem proposed by Barrus, Ferrara, Vandenbussche, and Wenger. Given a graph $H$ and an integer $t$, what is $\operatorname{sat}_{t}\left(n, \mathfrak{R}{(H)}\right)$, the minimum number of edges in a $t$-edge-coloured graph $G$ on $n$ vertices such that $G$ does not contain a rainbow copy of $H$, but adding to $G$ a new edge in any colour from $\{1,2,\ldots,t\}$ creates a rainbow copy of $H$? Here, we completely characterize the growth rates of $\operatorname{sat}_{t}\left(n, \mathfrak{R}{(H)}\right)$ as a function of $n$, for any graph $H$ belonging to a large class of connected graphs and for any $t\geq e(H)$. This classification includes all connected graphs of minimum degree $2$. In particular, we prove that $\operatorname{sat}_{t}\left(n, \mathfrak{R}{(K_r)}\right)=\Theta(n\log n)$, for any $r\geq 3$ and $t\geq {r \choose 2}$, thus resolving a conjecture of Barrus, Ferrara, Vandenbussche, and Wenger. We also pose several new problems and conjectures.

中文翻译:

图形的彩虹饱和度

在本文中,我们研究了 Barrus、Ferrara、Vandenbussche 和 Wenger 提出的以下问题。给定一个图 $H$ 和一个整数 $t$,$\operatorname{sat}_{t}\left(n, \mathfrak{R}{(H)}\right)$ 的最小边数是多少在 $t$-edge-coloured 图中 $G$ 在 $n$ 个顶点上,使得 $G$ 不包含 $H$ 的彩虹副本,但添加到 $G$ 的任何颜色的新边来自 $\{ 1,2,\ldots,t\}$ 创建了 $H$ 的彩虹副本?在这里,我们将 $\operatorname{sat}_{t}\left(n, \mathfrak{R}{(H)}\right)$ 的增长率完全表征为 $n$ 的函数,对于任何图 $ H$ 属于一大类连通图并且对于任何 $t\geq e(H)$。该分类包括所有最小度为 $2$ 的连通图。特别地,我们证明 $\operatorname{sat}_{t}\left(n, \mathfrak{R}{(K_r)}\right)=\Theta(n\log n)$,对于任何 $r\geq 3$ 和 $t\geq {r \choose 2}$,从而解决了 Barrus、Ferrara、Vandenbussche 和 Wenger 的猜想。我们还提出了几个新的问题和猜想。
更新日期:2019-12-20
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