On the ratios between packing and domination parameters of a graph

The relationship ρL(G) ≤ ρ(G) ≤ γ (G) between the lower packing number ρL(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 ir and IR, the lower and upper independence numbers i and β , and the lower and upper domination numbers γ and Γ ). In particular, best possible constants aθ , bθ , cθ and dθ are found for which the inequalities aθ θ(G) ≤ ρL(G) ≤ bθ θ(G) and cθ θ(G) ≤ ρ(G) ≤ dθ θ(G) hold for any connected graph G and all θ ∈ {ir, γ , i, β,Γ , IR}. From our work it follows, for example, that ρL(G) ≤ 2 ir(G) and ρ(G) ≤ 3 2 ir(G) for any connected graph G, and that these inequalities are best possible. © 2008 Elsevier B.V. All rights reserved.