最小度至少为n/q的大图中所有阶数的2-连通子图
$k$-Connected Subgraphs of All Orders in Large Graphs with Minimum Degree at Least $n/q$
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中文总结 AI 辅助
研究证实刘和宁猜想，即最小度至少为n/q的2-连通图含各阶2-连通子图。通过构造子图D及添加短路径，结合平均论证得到结果，还提出关于r-连通图的类似猜想。
中文摘要 AI 辅助
我们证实了刘和宁的一个猜想：对于每个固定整数q≥3，存在一个整数n_0(q)，使得每个阶数n≥n_0(q)且最小度δ(G)≥n/q的2-连通图G，对于每个ℓ∈{4,5,…,n}都包含一个阶数为ℓ的2-连通子图。证明有两个主要部分。首先，构造一个小的2-连通子图D，使D外每个顶点在D中有至少两个邻居，通过连续添加短路径得到从|V(D)|到n的每个阶数的2-连通子图。其次，对公共邻域的平均论证产生一个大的完全二部子图K_2,t，它提供了所有其余阶数的2-连通子图。我们还提出一个猜想：对于每对固定整数r≥2和q≥3，每个足够大阶数n且δ(G)≥n/q的r-连通图G包含从2r到n的每个阶数的r-连通子图。
英文摘要
For every fixed pair of integers $k\ge 2$ and $q\ge 3$, we prove that every sufficiently large $k$-connected graph $G$ of order $n$ with minimum degree $δ(G)\ge n/q$ contains a $k$-connected subgraph of every order $\ell\in\{2k,2k+1,\ldots,n\}$. In the case $k=2$, this confirms a conjecture of Liu and Ning~\cite{LiuNing}. The proof combines two constructions. First, we construct a small $k$-connected subgraph $D$ such that every vertex outside $D$ has at least $k$ neighbors in $D$. By successively adding the vertices outside $D$, we obtain $k$-connected subgraphs of every order from $|V(D)|$ to $n$. Second, an averaging argument on common neighborhoods, together with a complete bipartite construction, yields $k$-connected subgraphs of every order from $2k$ to $|V(D)|$. Together, the two constructions cover all orders from $2k$ to $n$. The lower endpoint $2k$ is best possible.
