路径、星型图与环的弱彩虹饱和数
Weak rainbow saturation numbers of paths, stars and cycles
- Nankai University（南开大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文精确确定了路径与星型图的弱彩虹饱和数，证明其分别随阶数线性和二次增长，并通过显式构造否定了环的弱彩虹饱和数首项为3n/2的猜想，给出更优的上界。
AI中文摘要：
若一个边着色图的所有边颜色均不相同，则称其为“彩虹图”。对于固定图$H$，若边着色图$F$的补图边集$E(\bar{F})$存在一种排序$e_1,e_2,\ldots,e_{|E(\bar{F})|}$，使得对$E(\bar{F})$的任意满足$c(e_i)\neq c(e_j)$的边着色$c$，在图$F+\{e_1,e_2,\ldots,e_i\}$中总能找到包含边$e_i$的$H$的彩虹拷贝，则称$F$是弱$H$-彩虹饱和的。弱彩虹饱和数$\operatorname{rwsat}(n,H)$定义为$n$个顶点的弱$H$-彩虹饱和图的最小边数。Li、Ma和Xie[JGT, 2025]证明，对任意非空图$H$，极限$\lim_{n\to\infty} \frac{\operatorname{rwsat}(n,H)}{n}$均存在。\n在同阶所有树中，路径和星型图分别取得普通弱饱和数的最小值和最大值，本文精确确定了二者的弱彩虹饱和数。对所有$\ell>30$，我们证明$\ell+1=\operatorname{rwsat}(n,P_\ell)< \operatorname{rwsat}(n,S_\ell)=\binom{\ell}{2}-1$，其中$P_\ell$和$S_\ell$分别表示$\ell$个顶点的路径和星型图。由此可知，路径的弱彩虹饱和数随$\ell$线性增长，星型图则呈二次增长。随后我们聚焦于环的情况，Li、Ma和Xie曾提出疑问：对任意$\ell\ge4$，$\operatorname{rwsat}(n,C_\ell)$的首项是否为$\frac32n$？我们通过显式构造给出否定回答，证明对任意$\ell\ge4$和所有足够大的$n$，有$\operatorname{rwsat}(n,C_\ell)< \frac{\ell}{\ell-1}n+c_\ell$，其中$c_\ell$仅依赖于$\ell$。由于$\frac{\ell}{\ell-1}<\frac32$，该结果严格改进了所有$\ell\ge4$的环$C_\ell$对应的首项系数猜想。
英文摘要：
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 $\ell\ge4$.
