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Keywords:
extremal problems; local property; locally tree; $k$-tree
Summary:
A graph $G$ is a {\it locally $k$-tree graph} if for any vertex $v$ the subgraph induced by the neighbours of $v$ is a $k$-tree, $k\ge 0$, where $0$-tree is an edgeless graph, $1$-tree is a tree. We characterize the minimum-size locally $k$-trees with $n$ vertices. The minimum-size connected locally $k$-trees are simply $(k+1)$-trees. For $k\ge 1$, we construct locally $k$-trees which are maximal with respect to the spanning subgraph relation. Consequently, the number of edges in an $n$-vertex locally $k$-tree graph is between $\Omega (n)$ and $O(n^2)$, where both bounds are asymptotically tight. In contrast, the number of edges in an $n$-vertex $k$-tree is always linear in $n$.
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