发表机构
School of Mathematics and Statistics, Fuzhou University（福州大学数学与统计学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文针对至少含一条边的固定有限简单图$F$，证明了$k$均匀超图的Berge-$F$-饱和数$\text{sat}_k(n,\text{Berge-}F)$为$O_{F,k}(n)$，即存在线性上界，解决了相关猜想的单图情形。
AI 中文摘要
对于至少含一条边的图$F$，若$k$均匀超图不含Berge-$F$，但添加任意缺失超边后会产生Berge-$F$，则称其为Berge-$F$-饱和超图。饱和数$\text{sat}_k(n,\text{Berge-}F)$是$n$个顶点的此类超图的最小边数。English、Gordon、Graber、Methuku和Sullivan猜想，对每个固定有限禁图族，该饱和数存在线性上界。本文证明了单图情形：对每个固定整数$k\boldsymbol{\text{≥}}2$和每个至少含一条边的固定有限简单图$F$，有$\text{sat}_k(n,\text{Berge-}F)=O_{F,k}(n)$。该证明结合了稀疏构造与度、匹配论证。
英文摘要
For a graph $F$ with at least one edge, a $k$-uniform hypergraph is Berge-$F$-saturated if it contains no Berge-$F$, but adding any missing hyperedge creates a Berge-$F$. The saturation number $\text{sat}_k(n,\text{Berge-}F)$ is the minimum number of edges in such a hypergraph on $n$ vertices. English, Gordon, Graber, Methuku and Sullivan conjectured a linear upper bound for every fixed finite family of forbidden graphs. We prove the single-graph case: $\text{sat}_k(n,\text{Berge-}F)=O_{F,k}(n)$ for every fixed integer $k\ge2$ and every fixed finite simple graph $F$ with at least one edge. The proof combines a sparse construction with degree and matching arguments.
Comments9 pages, 1 figures
