Geodesic
On extremal sizes of locally $k$-tree graphs
Czechoslovak Mathematical Journal, Tome 60 (2010) no. 2, pp. 571-587.

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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 $n$.
Classification : 05C05, 05C35
Keywords: extremal problems; local property; locally tree; $k$-tree 
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     title = {On extremal sizes of locally $k$-tree graphs},
     journal = {Czechoslovak Mathematical Journal},
     pages = {571--587},
     publisher = {mathdoc},
     volume = {60},
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     mrnumber = {2657970},
     zbl = {1224.05246},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/CMJ_2010__60_2_a20/}
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Borowiecki, Mieczysław; Borowiecki, Piotr; Sidorowicz, Elżbieta; Skupień, Zdzisław. On extremal sizes of locally $k$-tree graphs. Czechoslovak Mathematical Journal, Tome 60 (2010) no. 2, pp. 571-587. http://geodesic.mathdoc.fr/item/CMJ_2010__60_2_a20/