Rainbow saturation of graphs

TL;DR

This paper characterizes the minimum edges in edge-colored graphs avoiding rainbow subgraphs of a given graph, revealing growth rates and resolving a conjecture for complete graphs.

Contribution

It provides a complete classification of rainbow saturation numbers for a broad class of graphs, including a proof that for complete graphs, the saturation number grows as Theta(n log n).

Findings

Saturation numbers are Theta(n log n) for complete graphs K_r with r ≥ 3.

The classification applies to all connected graphs with minimum degree 2.

The paper resolves a conjecture regarding rainbow saturation for complete graphs.

Abstract

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.