关于具有最少边数的彩虹饱和图
On rainbow saturated graphs with minimum number of edges
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中文总结 AI 辅助
本文证明彩虹饱和数二分法：含孤立边的图饱和数为常数，否则为线性；并确定广义友谊图的渐近精确值。
中文摘要 AI 辅助
设 $F$ 是一个没有孤立顶点的固定图。一个边染色图是 $F$-彩虹饱和的，如果它不包含 $F$ 的彩虹副本，但添加任何缺失的边（以任意颜色）都会产生 $F$ 的彩虹副本。彩虹饱和数 $rsat(n,F)$ 是在 $n$ 个顶点上这样的图的最小边数。我们证明了一个由孤立边支配的二分法：如果 $F$ 包含一条孤立边，那么对于所有足够大的 $n$，$rsat(n,F)=O(1)$；而如果 $F$ 没有孤立边，那么 $rsat(n,F)=\Theta(n)$。线性下界用有向权重参数 $\eta(F)$ 表示，并确立了二分法的线性一半；在若干情况下，它也加强了 Cameron--Puleo 型系数。对于有界的一半，我们为形如 $H\cup K_2$ 的目标构造了彩虹饱和图。作为这些构造的一个应用，我们确定了广义友谊图 $F_{t,p,q}=tK_p\vee K_q$ 的彩虹饱和数的渐近精确行为，证明了当 $t\geq 2$、$p\geq 2$ 和 $q\geq 1$ 固定且 $n\to\infty$ 时，$rsat(n,F_{t,p,q})=(p+q-1)n+O(1)$。
英文摘要
Let $F$ be a fixed graph without isolated vertices. An edge-colored graph is $F$-rainbow saturated if it contains no rainbow copy of $F$, but the addition of any missing edge in any color creates a rainbow copy of $F$. The rainbow saturation number $rsat(n,F)$ is the minimum number of edges in such a graph on $n$ vertices. We prove a dichotomy governed by isolated edges: if $F$ contains an isolated edge, then $rsat(n,F)=O(1)$ for all sufficiently large $n$, while if $F$ has no isolated edge, then $rsat(n,F)=Θ(n)$. The linear lower bound is expressed in terms of a directed weight parameter $η(F)$ and establishes the linear half of the dichotomy; in several cases it also strengthens the Cameron--Puleo type coefficient. For the bounded half, we construct rainbow saturated graphs for targets of the form $H\cup K_2$. As an application of these constructions, we determine the asymptotically tight behavior for the rainbow saturation number of the generalized friendship graph $F_{t,p,q}=tK_p\vee K_q$, proving that $ rsat(n,F_{t,p,q})=(p+q-1)n+O(1)$ for fixed $t\geq 2$, $p\geq 2$ and $q\geq 1$ as $n\to\infty$.
发表机构
- Tsinghua University（清华大学）
机构由 AI 辅助整理，请以论文原文为准。
