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

Abstract

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 $\delta(G)\ge n/q$ 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.