Open packing, total domination, and the -Radon number

Let r ( G ) , ρ o ( G ) , and γ t ( G ) denote the P 3 -Radon number, the open packing number, and the total domination number of a graph G . We prove that r ( T ) ≤ 2 ρ o ( T ) + 1 for every tree T and r ( G ) < 2 γ t ( G ) + 1 for every non-trivial regular graph G .