On the ratios between packing and domination parameters of a graph

The relationship ρL(G)≤ρ(G)≤γ(G)ρL(G)≤ρ(G)≤γ(G) between the lower packing number ρL(G)ρL(G), the packing number ρ(G)ρ(G) and the domination number γ(G)γ(G) of a graph GG 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 irir and IRIR, the lower and upper independence numbers ii and ββ, and the lower and upper domination numbers γγ and ΓΓ). In particular, best possible constants aθaθ, bθbθ, cθcθ and dθdθ are found for which the inequalities aθθ(G)≤ρL(G)≤bθθ(G) and cθθ(G)≤ρ(G)≤dθθ(G) hold for any connected graph GG and all θ∈{ir,γ,i,β,Γ,IR}θ∈{ir,γ,i,β,Γ,IR}. From our work it follows, for example, that ρL(G)≤32ir(G) and ρ(G)≤32ir(G) for any connected graph GG, and that these inequalities are best possible.