LetGbe a graph and letvbe a vertex ofG. The open neigbourhoodN(v)ofvis the set of all vertices adjacent withvinG. An open packing ofGis a set of vertices whose open neighbourhoods are pairwise disjoint. The lower open packing number ofG, denoted#x3C1;#xB0;L(G), is the minimum cardinality of a maximal open packing ofGwhile the (upper) open packing number ofG, denoted#x3C1;#xB0;(G), is the maximum cardinality among all open packings ofG. It is known (see 7) that ifGis a connected graph of order n #x2265;3, then#x3C1;#xB0;(G)#x2264; 2n/3 and this bound is sharp (even for trees). As a consequence of this result, we know that#x3C1;#xB0;L(G)#x2264; 2n/3. In this paper, we improve this bound whenGis a tree. We show that ifGis a tree of ordernwith radius 3, then#x3C1;#xB0;L(G)#x2264;n/2 + 2 #x221A;n-1, and this bound is sharp, while ifGis a tree of ordernwith radius at least 4, then#x3C1;#xB0;L(G)is bounded above by 2n/3#x2014;O#x221A;n).
展开▼
