Packings in bipartite prisms and hypercubes.

The 2-packing number ρ 2 (G) of a graph G is the cardinality of a largest 2-packing of G and the open packing number ρ o (G) is the cardinality of a largest open packing of G , where an open packing (resp. 2-packing) is a set of vertices in G no two (closed) neighborhoods of which intersect. It is proved that if G is bipartite, then ρ o (G □ K 2) = 2 ρ 2 (G). For hypercubes, the lower bounds ρ 2 (Q n) ≥ 2 n − ⌊ log ⁡ n ⌋ − 1 and ρ o (Q n) ≥ 2 n − ⌊ log ⁡ (n − 1) ⌋ − 1 are established. These findings are applied to injective colorings of hypercubes. In particular, it is demonstrated that Q 9 is the smallest hypercube which is not perfect injectively colorable. It is also proved that γ t (Q 2 k × H) = 2 2 k − k γ t (H) , where H is an arbitrary graph with no isolated vertices. [ABSTRACT FROM AUTHOR]