Domination in partitioned graphs

Let V1, V2 be a partition of the vertex set in a graph G, and let γi denote the least number of vertices needed in G to dominate Vi. We prove that γ1 + γ2 ≤ 45 |V (G)| for any graph without isolated vertices or edges, and that equality occurs precisely if G consists of disjoint 5-paths and edges between their centers. We also give upper and lower bounds on γ1 + γ2 for graphs with minimum valency δ, and conjecture that γ1 + γ2 ≤ 4 δ+3 |V (G)| for δ ≤ 5. As δ gets large, however, the largest possible value of (γ1 + γ2)/|V (G)| is shown to grow with the order of log δ δ .