On the ratios between packing and domination parameters of a graph

The relationship ρ L ( G ) ≤ ρ ( G ) ≤ γ ( G ) between the lower packing number ρ L ( G ) , the packing number ρ ( G ) and the domination number γ ( G ) of a graph G is well known. In this paper we establish best possible bounds on the ratios of the packing numbers of any (connected) graph to its six domination-related parameters (the lower and upper irredundance numbers i r and I R , the lower and upper independence numbers i and β , and the lower and upper domination numbers γ and Γ ). In particular, best possible constants a θ , b θ , c θ and d θ are found for which the inequalities a θ θ ( G ) ≤ ρ L ( G ) ≤ b θ θ ( G ) and c θ θ ( G ) ≤ ρ ( G ) ≤ d θ θ ( G ) hold for any connected graph G and all θ ∈ { i r , γ , i , β , Γ , I R } . From our work it follows, for example, that ρ L ( G ) ≤ 3 2 i r ( G ) and ρ ( G ) ≤ 3 2 i r ( G ) for any connected graph G , and that these inequalities are best possible.