Dominating sets, packings, and the maximum degree

The inequality ρ ( G ) ≤ γ ( G ) between the packing number ρ ( G ) and the domination number γ ( G ) of a graph G is well known. For general graphs G , there exists no upper bound on γ ( G ) of the form γ ( G ) ≤ f ( ρ ( G ) ) where f is a function, as is remarked in [Discrete Math. 309 (2009), 2473–2478]. In this paper, we observe that γ ( G ) ≤ Δ ( G ) ρ ( G ) , where Δ ( G ) denotes the maximum degree of G . We characterize connected graph G with Δ ( G ) ≤ 3 that achieve equality in this bound. We conjecture that if G is a connected graph with Δ ( G ) ≤ 3 , then γ ( G ) ≤ 2 ρ ( G ) , with the exception of three graphs, one of which is the Petersen graph. We verify this conjecture in the case of claw-free graphs.