2-Connected Subgraphs of All Orders in Large Graphs with Minimum Degree at Least n/q

We confirm a conjecture of Liu and Ning\cite{LiuNing}: for every fixed integer $q\ge 3$, there exists an integer $n_0(q)$ such that every $2$-connected graph $G$ of order $n\ge n_0(q)$ with minimum degree $\delta(G)\ge n/q$ contains a $2$-connected subgraph of order $\ell$ for each $\ell\in\{4,5,\ldots,n\}$. The proof has two main components. First, we construct a small $2$-connected subgraph $D$ such that every vertex outside $D$ has at least two neighbors in $D$. By successively adding short paths, we obtain $2$-connected subgraphs of every order from $|V(D)|$ to $n$. Second, an averaging argument on common neighborhoods yields a large complete bipartite subgraph $K_{2,t}$, which provides $2$-connected subgraphs of all the remaining orders.Finally, we propose the following conjecture: for every pair of fixed integers $r\ge 2$ and $q\ge 3$, every $r$-connected graph $G$ of sufficiently large order $n$ with $\delta(G)\ge n/q$ contains an $r$-connected subgraph of every order from $2r$ to $n$.