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 $\delta (G)\ge n\mathrm{/}q$ contains a $2$-connected subgraph of order $\mathrm{\ell }$ for each $\mathrm{\ell }\in \{4,5,\dots ,n\}$. 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 the vertices outside $D$, we obtain $2$-connected subgraphs of every order from $\mathrm{\mid }V(D)\mathrm{\mid }$ 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 $\delta (G)\ge n\mathrm{/}q$ contains an $r$-connected subgraph of every order from $2r$ to $n$.