关于三个团的并的饱和数的注记
A note on the saturation number for unions of three cliques
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中文总结 AI 辅助
本文针对2≤p≤q<r<p+q且n足够大的情况，确定了三个团K_p、K_q、K_r的并的饱和数及唯一极图，结合已有结果完成了所有足够大n下该类并的饱和数及极图的确定。
中文摘要 AI 辅助
若图G不含F的复制，但对G的每条缺失边e，添加边e后G+e含F的复制，则称G是F-饱和图。饱和数sat(n,F)是n顶点F-饱和图的最小边数。受Faudree、Ferrara、Gould和Jacobson提出的关于K_p∪K_q∪K_{q+1}问题的启发，当2≤p≤q<r<p+q且n足够大时，本文确定了K_p∪K_q∪K_r的饱和数及唯一极图。结合已知的r≥p+q和r=q时的结果，本文完成了所有足够大n下三个团的并的饱和数及极图的确定。
英文摘要
A graph $G$ is $F$-saturated if $G$ contains no copy of $F$ but $G+e$ contains a copy of $F$ for every missing edge $e$ of $G$. The saturation number $\sat(n,F)$ is the minimum number of edges in an $n$-vertex $F$-saturated graph. Motivated by a problem posed by Faudree, Ferrara, Gould, and Jacobson concerning $K_p\cup K_q\cup K_{q+1}$, we determine the saturation number and the unique extremal graph for $K_p\cup K_q\cup K_r$ whenever $2\le p\le q<r<p+q$ and $n$ is sufficiently large. Together with the previously known results for $r\ge p+q$ and for $r=q$, this completes the determination of the saturation number and the extremal graphs for unions of three cliques, for all sufficiently large $n$.
