Weak rainbow saturation numbers of paths, stars and cycles

Abstract

An edge-colored graph is \emph{rainbow} if all of its edges receive distinct colors. For a fixed graph $H$, an edge-colored graph $F$ is called weakly $H$-rainbow saturated if there exists an ordering $e_1,e_2,\ldots,e_{|E(\bar{F})|}$ of $E(\bar{F})$ such that, for any edge coloring $c$ of $E(\bar{F})$ with $c(e_i)\neq c(e_j)$, there is always a rainbow copy of $H$ that contains $e_i$ in $F+\{e_1,e_2,\ldots,e_i\}$. The \emph{weak rainbow saturation number} $\operatorname{rwsat}(n,H)$ is the minimum number of edges in a weakly $H$-rainbow saturated graph on $n$ vertices. Li, Ma, and Xie [JGT, 2025] showed that $\lim_{n\to\infty} \frac{\operatorname{rwsat}(n,H)}{n}$ exists for every nonempty graph $H$. Paths and stars attain, respectively, the minimum and maximum ordinary weak saturation numbers among all trees of the same order. We determine their weak rainbow saturation numbers exactly. For all $\ell>30$, we show that $$ \ell+1=\s(n,P_\ell)<\s(n,S_\ell)=\binom{\ell}{2}-1$$ where $P_\ell$ and $S_\ell$ denote the path and star on $\ell$ vertices, respectively. Thus, their dependence on $\ell$ is linear for paths and quadratic for stars. We then focus on cycles. Li, Ma, and Xie asked whether $\operatorname{rwsat}(n,C_\ell)$ has leading term $\frac32n$ for every $\ell\ge4$. We answer this question negatively by giving an explicit construction showing that, for every $\ell\ge4$ and all sufficiently large $n$, $$\s(n,C_\ell)<\frac{\ell}{\ell-1}n+c_\ell,$$ where $c_\ell$ depends only on $\ell$. Since $\frac{\ell}{\ell-1}<\frac32$, this strictly improves the proposed leading coefficient for every cycle $C_\ell$ with $\ell\ge4$.