On extremal sizes of locally k-tree graphs

Mieczysław Borowiecki; Piotr Borowiecki; Elżbieta Sidorowicz; Zdzisław Skupień

doi:10.1007/s10587-010-0037-z

On extremal sizes of locally k-tree graphs

Abstract

A graph G is a locally k-tree graph if for any vertex v the subgraph induced by the neighbours of v is a k-tree, k>=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 >= 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 boundsare asymptotically tight. In contrast, the number of edges in an n-vertex k-tree is always linear in n.

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Category:: Articles
Type:: artykuł w czasopiśmie wyróżnionym w JCR
Published in:: CZECHOSLOVAK MATHEMATICAL JOURNAL no. 60, pages 571 - 587,
ISSN: 0011-4642
Language:: English
Publication year:: 2010
Bibliographic description:: Borowiecki M., Borowiecki P., Sidorowicz E., Skupień Z.: On extremal sizes of locally k-tree graphs// CZECHOSLOVAK MATHEMATICAL JOURNAL. -Vol. 60, nr. Iss. 2 (2010), s.571-587
DOI:: Digital Object Identifier (open in new tab) 10.1007/s10587-010-0037-z
Verified by:: Gdańsk University of Technology