## Abstract

Let G be a graph and let v be a vertex of G. The open neigbourhood N(v) of v is the set of all vertices adjacent with v in G. An open packing of G is a set of vertices whose open neighbourhoods are pairwise disjoint. The lower open packing number of G, denoted ρ^{°}_{L}(G), is the minimum cardinality of a maximal open packing of G while the (upper) open packing number of G, denoted ρ^{°}(G), is the maximum cardinality among all open packings of G. It is known (see [7]) that if G is a connected graph of order n ≥3, then ρ^{°}(G) ≤ 2n/3 and this bound is sharp (even for trees). As a consequence of this result, we know that ρ^{°}_{L}(G) ≤ 2n/3. In this paper, we improve this bound when G is a tree. We show that if G is a tree of order n with radius 3, then ρ^{°}_{L}(G) ≤ n/2 + 2 √n-1, and this bound is sharp, while if G is a tree of order n with radius at least 4, then ρ^{°}_{L}(G) is bounded above by 2n/3—O√n).

Original language | English |
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Pages (from-to) | 235-245 |

Number of pages | 11 |

Journal | Quaestiones Mathematicae |

Volume | 21 |

Issue number | 3-4 |

DOIs | |

Publication status | Published - Nov 1998 |

Externally published | Yes |

## ASJC Scopus subject areas

- Mathematics (miscellaneous)