Weak rainbow saturation numbers of graphs
Xihe Li, Jie Ma, Tianying Xie
TL;DR
We address the weak rainbow saturation problem for a fixed graph $H$ and establish that the linear growth rate ${\rm rwsat}(n,H)$ has a well-defined asymptotic density: ${\lim_{n\to\infty}}{\rm rwsat}(n,H)/n$ exists for every non-empty $H$. The authors develop a subadditive framework and a series of lemmas to glue together smaller weakly rainbow saturated graphs while preserving the property, then apply Fekete's lemma to obtain the limit. They provide lower and upper bounds in terms of $\delta'(H)$ and a parameter $f(H)$, showing the limit is zero precisely when $H$ contains a pendant edge, and strictly positive otherwise. The results generalize prior work on rainbow saturation and resolve an open question of Behague et al., offering a unified picture for the asymptotics of weak rainbow saturation across all non-empty graphs, including implications for cycles and complete graphs.
Abstract
For a fixed graph $H$, we say that an edge-colored graph $G$ is \emph{weakly $H$-rainbow saturated} if there exists an ordering $e_1, e_2, \ldots, e_m$ of $E\left(\overline{G}\right)$ such that, for any list $c_1, c_2, \ldots, c_m$ of pairwise distinct colors from $\mathbb{N}$, the non-edges $e_i$ in color $c_i$ can be added to $G$, one at a time, so that every added edge creates a new rainbow copy of $H$. The \emph{weak rainbow saturation number} of $H$, denoted by $rwsat(n,H)$, is the minimum number of edges in a weakly $H$-rainbow saturated graph on $n$ vertices. In this paper, we show that for any non-empty graph $H$, the limit $\lim_{n\to \infty} \frac{rwsat(n, H)}{n}$ exists. This answers a question of Behague, Johnston, Letzter, Morrison and Ogden [{\it SIAM J. Discrete Math.} (2023)]. We also provide lower and upper bounds on this limit, and in particular, we show that this limit is nonzero if and only if $H$ contains no pendant edges.