Open packing, total domination, and the P3P3-Radon number

Let r(G)r(G), ρo(G)ρo(G), and γt(G)γt(G) denote the P3P3-Radon number, the open packing number, and the total domination number of a graph GG. We prove that r(T)≤2ρo(T)+1r(T)≤2ρo(T)+1 for every tree TT and r(G)<2γt(G)+1r(G)<2γt(G)+1 for every non-trivial regular graph GG.