On the number of 2-packings in a connected graph

By a 2-packing in a graph we mean a subset of its vertex set, in which all the vertices are in distance at least 3 from each other. estion about the maximum number of 2-packings in a graph is strongly related to the problem of L ( 2 , 1 ) -labeling of graphs. s paper we find new asymptotic upper and lower bounds on the maximum number of 2-packings in a connected graph on n vertices. The bounds are O ( 1.5399 … n ) and Ω ( 1.4970 … n ) , respectively. er, we present a lower bound on the number of k -element 2-packings in a connected graph, which is max ( n − 2 k + 2 k , ( k + 1 ) ⌊ n − 1 2 ⌋ k ) .